Can you plug a pre-amp into a pre-amp?

Sure. Run your table into the preamp that has the phono stage and then use either the preamp output or the tape output of that preamp to any line level input on the other preamp. If you use the preamp output, you can trim the levels with the voulme control. If you use the tape outputs, you will have a fixed output, but it will be a cleaner signal.

I have done this many times.

+1 I've done it too. It's a great way to use a good quality vintage phono preamp and a great remote controllable (or not) new line stage.

Yes, I do it with chase KP1 plugged into a Kenwood C1. The kp1 gives me remote control for the C1 which doesn't have that. I love the KP1 since it also has balance and tone controls. The KP1's come up on ebay often and go for about $100, try one, you will be surprised in that it does not degrade the other component.