Bandspreading this receiver was one of the more difficult projects I've done, mostly because I used the variable capacitor that came with the signal generator. I hope my experience will help others avoid the same pitfall.
Tank Circuit Components For Bandspreading
The bandswitching/bandspreading method I used is much simpler than the plug-in coil method used in the famous National SW-3 (1931). The next two schematics are taken from a drawing done by W2AMI.
National SW-3 Detector Stage With Bandspread Coil
Bandspread And General-Coverage Coils For The National SW-3
Unfortunately, only after I'd finished the project did I discover a 1977 article that might have made the project even easier.  Instead, I made a spreadsheet to help me select the required series and parallel capacitors. To determine the starting inductance for each band, I set the desired mid-band tank reactance at 100 ohms and selected the inductance based on that value. The 100-ohm value worked out well, though later I will suggest an experiment you may wish to try. 
You can download the spreadsheet and try it out. I wish I could say it will solve all your problems in selecting the capacitors. Stray capacitance and self-capacitance of the coil will, I'm afraid, make the spreadsheet only partially useful. Still, it's a start. If you enter the minimum and maximuum capacitance of your variable cap, and the coil inductance, you'll get an array of frequency ranges and reactances you can obtain with standard-value series capacitors. Chances are you'll need some parallel capacitance, too.
Using The Spreadsheet
This is how I used the spreadsheet. Inserting series capacitance (Cs) reduces the total capacitance, minimum and maximum. The formula for calculating the combination of two capacitors, Cs and Ct, in series is: (Cs * Ct) / (Cs + Ct). Let's say your tuning capacitor by itself covers 30-365 pF, a ratio of 12.2, and you place a 100-pF fixed capacitor in series with it.
(30 * 100) / (30 + 100) = 23.1 pF = Cmin
(365 * 100) / (365 + 100) = 78.5 pF = Cmax
78.5 / 23.1 = 3.4
So not only has Cs reduced the minimum and maximum capacitances, it has also reduced the differential between minimum and maximum capacitances. (The spreadsheet rounds off the ratio to zero decimal places.) How does this affect the tuning range?
Let's say we're building a 40-M receiver, and we want it to cover 7.0-7.1 MHz. Our variable capacitor by itself will resonate with a 1.42 µH coil at 7 MHz.
where f is in megahertz, L is in microhenries and C is in microfarads (pF/1,000,000).
Assuming inductance L is fixed, what will be the upper end of the tuning range? Wow! 24.8 MHz! Now, what if we use the series combination? We'll need a 6.6-µH coil to resonate at 7 MHz with variable capacitor Ct fully meshed. At minimuum capacity, though, the circuit resonates at 12.9 MHz. Still too high, but we're getting closer. Or are we?
Why 100 Ohms Reactance?
Earlier, I mentioned Dave Newkirk's suggestion of 100 pF as a good compromise on 40 meters. It's a compromise because higher reactances yield higher amplification in the regenerative stage. This feature was more important in the regenerative receiver's early days, when tube gains were low and tubes were expensive. Many receivers were battery operated, and you had to pay someone to recharge the A battery from which you powered the tube heaters. One tube added 10 or more watts of power consumption, so the more gain you could squeeze out of each stage the better. Today, FETs and bipolar transistors achieve gains far greater than those old tubes.
Why not get as much performance from the regenerative stage as possible? When the stage is operating at very high gain, it is very difficult to adjust the transition between just-barely oscillating and oscillation. If you've used a regen, you know that's the critical operating point. For modulated-carrier reception you want the circuit operating just below oscillation; for CW and SSB, just above. Besides that annoying "click" when the circuit jumps into oscillation, strong signals are more likely to "pull" the oscillator at higher gain levels. The 100-ohm reactance isn't carved in stone, and later I'll have a suggestion for determining the best reactance to use. For now, we'll use 100 ohms as a starting point.
The correct inductance to build a 100-ohm-reactance parallel-resonant tank circuit at 7 MHz is 4.5 µH. The capacitance that resonates with that inductance at 7 MHz is 115 pF. According to the spreadsheet, we need Cs = ~170 pF of series capacitance to bring our maximum capacitance down to 115 pF. Unfortunately, at minimum capacitance the circuit resonates way up at 14.9 MHz. What next?
At this point, I suggest forgetting about getting a series combination that alone will resonate over the desired frequency range. Instead, look for one that gives about the right range, regardless of frequency. Actually, what we want is the right ratio of upper to lower frequency. Column I in the spreadsheet provides this information for your variable capacitor, for a range of standard fixed-capacitor values. Note that column I doesn't change when you vary the inductance. For a given Ct, a given Cs will always provide the same ratio of upper to lower frequency. Now we have something to work with.
Our desired ratio is 7.1 / 7.0, or 1.0143. Not very large. Can we do it? Yes, but it won't be easy. I changed the value in cell A13 from 10 to 3, and got 1.04 in cell I13. (I know, the spreadsheet says to not change those values; the ones you really want to stay away from are the formulas in columns B-L.) Three pF is a pretty small capacitor, and stray capacitance in your circuit easily will exceed that. I ran into this very problem when building my receiver. The only solution was to accept wider tuning ranges on some bands. Let's assume for now that we can get away with such a small Cs. The tuning range at this point is 43.5-45.4 MHz. Now we can start adding Cp to bring the lower limit down to 7 MHz.
Can't Get There From Here
If you're following along with the spreadsheet, you should see the problem. Getting the desired upper-lower frequency ratio has made it impossible to resonate with L on 7 MHz. Ct is so large in relation to Cs, that even adding 100,000 pF in parallel with the coil only brings the frequency down to 43 MHz, and the ratio has dropped to 1.
If we can't get what we want with this variable capacitor, what can we get, without violently ripping off some of its plates? Let's lower our sights, and our lower frequency limit. Instead of starting at 7 MHz, let's start at 6.2 MHz. This covers some maritime frequencies, where Morse Code transmissions are still heard once a year. We'll keep L at 4.5 µH though, because we want that 100-ohm reactance at the frequency of most interest, 7 MHz.
A 10-pF Cs provides the 1.14 ratio; now to try adding Cp. Here's our next hurdle. When we add parallel capacitance, we change the ratio. A parallel capacitance has a greater effect on the low-C end of the combination than the high end, where it is proportionally smaller. Just the opposite effect of adding series capacitance. (Yes, this is tedious, but easier than littering your bench with capacitors and breathing all those solder fumes.) And, yes, there are formulas for calculating these things, in reference 2. I'm not sure they'd be any easier to use, unless you wrote them into a spreadsheet.
Neither the spreadsheet or reference 2 will provide the answer we're looking for anyway, a combination of series and parallel capacitors of standard values, that lets us tune the exact range in which we're interested. In the end, I wound up using trimmer capacitors in almost every location, sometimes in combination with fixed capacitors. And don't forget stray capacitance, which is especially troublesome when you're working with large variable capacitors, as the series capacitance has to be so small. I a schematic of the finished receiver, without specifying the capacitors I used, because I didn't want to remove them for measuring! The coils were wound to provide 100-ohms reactance at the bottom edge of each ham band (17 meters on Band E, which also covers 15 meters.) This table shows the tuning ranges in kHz I settled for. (Yes, Band E needs tweaking, but this is a result of using a large capacitor on a high frequency: the series capacitor is quite small.)
BAND FROM TO A 3496 3608 B 6950 7074 C 9960 10,534 D 13,935 14,088 E 17,686 22,220
Finally, here are some suggestions. First, don't try this at home. No, I am certainly not a professional, or I wouldn't have tried it either. Trying to adapt a large variable capacitor for such applications is too hard. Find a smaller capacitor. If you haven't deleted the spreadsheet in disgust yet, try a 3-30 pF variable capacitor, and our 4.5-µH coil in parallel with 220 pF. The series capacitor is also 220 pF, for a tuning range of 6.94-7.13 MHz. With that much series C, stray capacitance around the switch is less of a problem, but be ready for its effects just the same. The same variable capacitor with 82 pF in series, and a 2.3-µH coil in parallel with 150 pF, will neatly cover the entire 20-M band, including the data and SSB portions. Moving lower in frequency, the same variable capacitor, used with a 9-µH coil in parallel with 210 pF, and no series capacitor, will tune 3.4-3.6 MHz. If I ever do this again . . . .
My other suggestion is to not start with 100 ohms as the tank reactance. I found my circuit would oscillate when the reactance was about 75 ohms, and probably would oscillate at even lower reactances. Although the gain is slightly less, I make it up in the audio section. The receiver doesn't "pull in" even at the high end of each band, where reactance is greatest, and it does operate very smoothly. But at the lower reactances, it operates even more smoothly. Try 50 ohms reactance at the desired frequency, and see if your regenerative stage reliably starts oscillating at the low end of the band. And, if you have to make coverage much wider than the desired band, try to get the excess at the low end, not the high end. That should give you better performance everywhere. You never know what you might hear outside the ham bands, or whatever range you design for. On the other hand, if you can limit coverage so the reactance stays relatively constant, you won't have to adjust the REGENERATION control as much as you tune around.
I used this receiver in the Summer 2009 RadioBoard Homebrew Contest. I'm posting my log as the contest progresses. RadioBoard Summer 2009 Contest Log.
1. Anderson, "Bandspreading Techniques For Resonant Circuits," Ham Radio, February 1977, pp 46-51.
2. Newkirk, "Rediscovering the Mix-Goodman Band-Imaging Receiver."