I agree with elevate, $1000 would not really be enough for a good new kit. A mid range 5 piece kit (something like
a Yamaha Stage Custom or a Pearl Export) will run about $800. Plan on putting better batter heads on the kit (this adds $100).
The key is buying good cymbals - you can improve the sound of a mid range drum kit with better heads and good tuning - but cymbals are what they are. For the studio you need mid to upper range cymbals which will cost about $200 for hi-hats, $200 for a ride and about $125 for a decent crash - so you're looking at a minimum of $500-$600 just for good cymbals (something like Zildjian A's,
Sabian AA's or
Paiste Signature).
This brings your total cost to around $1500 (for a mid range kit made of Mahogany and/or basswood with the minimum cymbal set-up). If you choose a Birch or Maple kit (much better sounding) you're getting into $1500 - $2000 just for the drums - you still need to add $500 for cymbals.
Mahogany and basswood are not ideal for jazz. Even Birch which is a very good wood for recording is not best suited to jazz. A thin ply maple kit works best (Maple is "warmer" sounding). Jazz kits tend to have an 18" kick drum whereas rock tens to have a 22" (or even 24" kick. As elevate said a "fusion sized" kit with 10", 12" and 14" toms and a 20" kick would be the best compromise. A standard 5 1/2 by 14 snare would probably be best - I would suggest wood over steel.
If you buy used, make sure you have a knowledgeable drummer look at the kit (before buying) to make sure the lug threads are OK and that the shells are true round with good bearing edges,
I have found that the sad truth is many drummers don't take good care of their gear and when they do sell it, it's pretty beat up.
Naturally, you need mics and the proper gear to track the drum sounds - more $$$$. Lastly, keep in mind 50% of a good recorded drum sound is the room. If you don't have a good sounding room, you will be less than satisfied with the recorded sound of the drums.
I've read your posts and you sound like you've paid your dues - I'm sure you'll make the right decision.