I'm rather scared to try a tank that wasn't meant to be used with an air compressor. Really, just keep an eye out for another compressor. You might find a deal on a new 20+ gallon compressor. I mean those noisy 21 gallon HF compressors are under $200 all the time.
I recently rebuilt my compressor with a used tank. My 20 gallon Campbell Hausfeld compressor sprang a leak due to tank rust. I found a 26 gallon CH compressor listed as missing it's pump and transferred things over (safety check, verify the tank's safe working pressure is higher than you pump and your blow off valve). I spent $40 on the tank and perhaps another $20 on miscellaneous parts to complete the transfer.
I did look for a new, 20+ gallon tank only. They tend to cost just as much as a new compressor. If your objective is to go super cheap on this job, look for a used, messed up compressor for under $100. If you have a bit more to spend and a second 120V circuit in the garage, get a second compressor and then make a cross over hose so you can run the two at the same time. This will give you around 2X the pump flow rate and more tank size.
Best option, get a bigger compressor - certainly "best" ignores cost/space/install effort etc.