When representing doesn't work

Let's say you have JJ and the flop is rainbow K 8 3.

You're first to act so you bet half the pot or so. A reraise is an automatic fold, obviously, but what to do when he calls and the turn is another overcard? Just accept that you're beaten and check-fold?

depends alot on the player and the stage of the SnG. If the second over is an A then you're definately done with the hand but without a solid read check/fold seems good. Maby call a small bet on the river if he checks the turn.

If the next card is an ace, check-raise, baby. They'll put you on AK and lay down every time. OK, maybe not... I've done it successfully before though.

I made a guy lay down KK with QQ last week. maybe my proudest moment.

Need a touch more info.
Did you raise JJ preflop?
If yeah then bet the pot. You HAVE THAT king, or at least soyour tying represent. If you have balls maybe bet the turn too if another overcard, or just check it.
If you didnt raise preflop its a simple check imo Why possibly waste the chips.

My 2 cents:

If you can afford throwing in another turn bet without crippling your stack too much do so.
Checking is a sign of weakness and you may be bluffed out of a pot with the best hand, throwing in another bet may make him fold in case he does not have a strong hand.

However if putting more money in the pot will make you pot commited check/fold

if its a queen no check fold, if its a ace, push...no way he just calls preflop with a/k late in the tourny..