Discounting jacks a little bit (due to lack of 3-bet) his sets number about 7-8 combos. While this seems like a tiny range, the problem is that that's the primary range that he's playing for stacks with if you shove (a big overbet). He can't have that many flushdraws, because you have the ace. That leaves mostly KdQd. Would he flat Td9d? Maybe throw in KdTd and QdTd if we think he's defending those, but we're still looking at an all-in range that is weighted towards sets.
The rest of his range is probably top-pair type hands, some air that takes a stab and some medium PPs that raise for information or turn their hands into bluffs or whatever. In my experience, his range for betting the turn will be even more weighted towards sets than anything else. Once you call the flop, he's unlikely to bet 88 unimproved, and he's going to be careful about betting a lot of the other made hands, too. The only part of his range that gets bet 100% will be his sets unless he's super-tricky and decides to go for a double check-raise (so rare you can basically ignore the probability).
For that reason, I like calling the flop. If he bets again on the turn, consider your odds
and whether or not you can draw profitably. Because his range is going to be very value-heavy when he bets the turn, you can probably include quite a bit of implied odds at that point (it's very hard for people to check/fold a set on the river when a third flushcard hits). If he checks the turn, I think the decision between checking or betting is dictated by the card that comes but is going to be very close on average. As in, I'm probably betting half the cards and checking half of them, if I had to guess. I'd break it down like this:
Bet the flush (9 cards), a king or a queen (6 cards) and another jack (3 cards). There are two reasons to bet the jack once he checks it: Firstly because you can probably bet/fold at that point. If he check-raises a jack on the turn, you'll be drawing dead pretty often. But it also reduces the probability that he has a jack, and moves his range more towards "bluffs", some of which beat you or split (33, 44, 66-99, AQ, another AT). And since you called the flop you can definitely have a jack, making it even harder for him to get ballsy and check-raise bluff you. Also, if he DID check-raise a flushdraw, he's almost certainly going to call a turn bet, which is obviously good for you.
Check all the other cards. So I bet 18 cards and check behind 29 of them. I guess I was a bit off in my guesstimate of how often I bet or check the turn.
Note that I wouldn't bet an ace or a ten. Not only because I wouldn't really know what to do if he check-raises (while we're certainly not ahead on average when it happens, we're obviously not drawing dead), but also because there's little in his range that will call a bet that we beat, besides weaker flushdraws (of which there are few) and maybe a hand like KJ or QJ (if the turn is an ace, not a ten, obviously) that check/calls once, but not always twice. And while there'd certainly be value in getting calls from KJ or QJ, that's not a very big part of his range anyway since he's somewhat unlikely to check-raise the flop with those.
So... Yeah. If the turn is 6s and goes check/check, the river decision is going to be a little tricky as well, but I'd probably bet the same range as above for the river cards, except now I'd bet an ace for value. If he leads out when we river a ten or an ace, I'm calling.