10NL: Double barrel this turn?

[Supwithbates]

PokerStars No-Limit Hold'em, $0.10 BB (7 handed) Hand History Converter Tool (http://poker-tools.flopturnriver.com/Hand-Converter.php) from FlopTurnRiver.com (http://www.flopturnriver.com) (Format: 2+2 Forums)

Button ($1.85)
SB ($6.75)
BB ($3)
UTG ($10.05)
MP1 ($6.55)
Hero ($11.75)
CO ($10.15)

Preflop: Hero is MP2 with A/images/graemlins/heart.gif, J/images/graemlins/heart.gif.
<font color="#666666">2 folds</font>, <font color="#CC3333">Hero raises to $0.4</font>, <font color="#666666">3 folds</font>, BB calls $0.30.

Flop: ($0.85) 9/images/graemlins/diamond.gif, 7/images/graemlins/club.gif, 7/images/graemlins/heart.gif <font color="#0000FF">(2 players)</font>
BB checks, <font color="#CC3333">Hero bets $0.4</font>, BB calls $0.40.

Turn: ($1.65) 8/images/graemlins/club.gif <font color="#0000FF">(2 players)</font>
BB checks, <font color="#CC3333">Hero bets $0.7</font>

Villian is new to table, no reads/stats. His timing struck me as weak on the flop, I thought he might be trying to float with a low PP. Even if he's a calling station that can't laydown a weak hand, I probably have ten outs to improve and if he doesn't fold turn he won't fold river, so I am giving him small reverse implied odds.
Spew or NH?

[straightflush]

Without reads I would just check behind and catch a ten.

If you are going to double barrel I would bet more.

[lippy]

Check behind and improve. No reason in building this pot and this isn't a very dry board (which would call for a 2nd barrel).

[Supwithbates]

Would the answers change if Villian's stack was 10$ instead of 3$?
I really think there's a lot to exploiting reverse implied odds situations vs. a probably calling station.