If you want to jump more seriously, even if not necessarily right away, you’re really better off buying/leasing. Most lesson barns are tailored to the 2’6 and under crowd, with the expectation that once the rider progresses past that, they buy/lease their own horse. You might get lucky and get to ride some nice sale horses, and jump bigger, perhaps show, but then you’re still not getting to keep the horse that you’re gaining confidence and experience on.
I think with your stated goal of jumping more, you’re best off buying or leasing.
Your price range is pretty low for something with much experience, but I’d definitely prioritize Sound But Green over Well-trained But NQR. Since you’re comfortable with a green horse, and you’re willing to buy OTTBs and off-breeds, you do have a good chance of finding a quality greener horse with a good brain and sound body.
Leasing gives you the chance to be more picky, but you’re going to pay for that privilege. In H/J land, most horses lease for 1/3-1/2 of their sale price/value per year.
So, I’d either A: lease a schoolmaster for a year to gain confidence and experience with bigger jumps, and then buy a green horse the following year, applying the skills and mileage you gained with the lease horse.
Or B: just buy a nice OTTB, and take your time. If you’re comfortable with some quirks and young-n-dumb behavior, and you have a good coach who can put some solid schooling rides on the horse, and mentor you so that you’re making progress, this might be the best plan, though it does have you on the longer-term path to jumping more and bigger.
What you shouldn’t do is try to talk yourself in to buying a horse that isn’t sound in light work, doing less than your goals are.
