I think I played it correctly, that's why I'm asking if there's another way I could have played it. Should I fold to his reraise on the turn?

PokerStars Zoom No-Limit Hold'em, $0.05 BB (9 handed) - PokerStars Converter Tool from http://www.flopturnriver.com/

Button ($7.20)
SB ($3.07)
Hero (BB) ($4.60)
UTG ($4.48)
UTG+1 ($6.85)
MP1 ($4.21)
MP2 ($3.46)
MP3 ($4.39)
CO ($5.05)

Preflop: Hero is BB with 8, 10
2 folds, MP1 raises to $0.15, 1 fold, MP3 calls $0.15, 3 folds, Hero calls $0.10

Flop: ($0.47) K, 8, J (3 players)
Hero checks, MP1 checks, MP3 checks

Turn: ($0.47) 10 (3 players)
Hero bets $0.29, 1 fold, MP3 raises to $0.66, Hero calls $0.37

River: ($1.79) 6 (2 players)
Hero checks, MP3 bets $0.89, Hero calls $0.89

Total pot: $3.57 | Rake: $0.15