PokerStars No-Limit Hold'em, $0.02 BB (8 handed) FTR converter on zerodivide.cx

UTG ($5.18)
UTG+1 ($5.39)
MP1 ($5.36)
MP2 ($1.04)
CO ($1.91)
Button ($1.66)
Hero ($5.93)
BB ($5.24)

Preflop: Hero is SB with K, K.
3 folds, MP2 raises to $0.04, CO calls $0.04, 1 fold, Hero raises to $0.20, 1 fold, MP2 calls $0.16, CO folds.

Flop: ($0.46) J, 3, 9 (2 players)
Hero bets $0.30, MP2 calls $0.30.

Turn: ($1.06) A (2 players)
Hero bets $0.60, MP2 calls $0.54 (All-In).

River: ($2.20) Q (2 players, 1 all-in)

Final Pot: $2.20

I put him on probably the flush draw on the turn. Should i have guessed he had the A or should i have pushed the flop. I came very close to pushing this flop but thought it would be better to push a non turn.