I thought my amp had a bad tube and got a sovtek 12ax7 cause it's all the store carried. In the end my tubes were fine so I could stay with the existing JJ's. Anyways, now I'm left with a spare, which is good to have, but I am curious. Could I make a "preamp/overdrive" pedal with the tube to add the special grit a tube offers? This would be nice if it worked with bass, cause I ordered a GK mb200 amp and it's solid state would be cool to have a tube in the mix just to check it out.
look up the valvecaster pedal
it simple enough to be built in a few hours and uses a tube
Unless you already build pedals you are talking about taking your $10 tube and adding $30 worth of parts, plus whatever tools and skills you don't already have.

If you're already into building stuff and just using the AX7 for inspiration, I'd do something like the Matchless Hot Box. Schematics for that abound.
Ive done a lot of research on the valvecaster and 12ax7 tubes DO NOT work good. from all the forums ive read they sound terrible because they arent built for low voltages (12v). they are built for hundreds/thousands of volts.

So there is no OD/Pre-amp stompbox that you can make with your 12ax7 (unless you want something mains powered, which I would leave to people who are qualified/confident enough to attempt them).

12au7 tubes are built specifically for low voltage applications, so if you do want a valvecaster, they are the way to go. also, the valvecaster is more of a boost stompbox than a OD/Pre-amp.