First, this hand is fairly important to the main hand.

Everleaf $0.01/$0.02 No Limit Hold'em - 4 players - View hand 1411432
DeucesCracked Poker Videos Hand History Converter

SB: $1.44
Hero (BB): $2.21
CO: $3.61
BTN: $2.02

Pre Flop: ($0.03) Hero is BB with A K
CO raises to $0.06, 1 fold, SB raises to $0.20, Hero raises to $0.54, 2 folds

Got it?
This is 3 hands later. 4 hands later. BB is 39/35 with a 1/4 3bet and 1/2 fold to steal. Pre is pretty standard I think, willing to AIPF with JJ 3 handed. Then the flop comes and it's pretty bad. Extremely bad. What's our plan?


Everleaf $0.01/$0.02 No Limit Hold'em - 3 players - View hand 1411434
DeucesCracked Poker Videos Hand History Converter

Hero (SB): $2.44
BB: $3.59
BTN: $1.98

Pre Flop: ($0.03) Hero is SB with J J
1 fold, Hero raises to $0.06, BB raises to $0.20, Hero raises to $0.60, BB calls $0.40

Flop: ($1.20) T A K (2 players)
Hero ?

C/F sunds good.  Given villain, I don't see firing here as doing much you can't accomplish at showdown, save freeze him firing off a few barrels with air.  I doubt you're getting any value with a bet OTF, or folding better.