I wonder if when you say "3 betting" what you really mean is opening the pot for 3x the BB. To me, 3 betting means somebody else raised and you re-raised. a re-raise is a 3 bet. if somebody re-re-raised it would be a 4bet.
anyways....the add-on is usually mandatory unless you have massively increased your stack by like 4x or more.
As for the hand in question: there are times to play A6 in early position, but not a lot of 'em.
Tournament poker is really all about stack size management. The size of your stack relative to the blinds and relative to the average stack determines which moves are in your arsenal, and which cards are in your range in which positions.
ex: If you're UTG with 1,000 and blinds are 400/800 then you have almost no options. You should basically shove any 2 cards every time.
ex: If you're UTG with 45,000 and blinds are 400/800 then basically every possible move is in your arsenal, and lots of hands are in your late position range if you are a skilled position player.
with a stack of 10,000 and blinds of 400/800 your options or moves have decreased, but you still do have a few moves in your arsenal. For me, I'd allow myself these particular moves:
open-shove (your stack is almost too big for the open shove though)
re-steal (somebody opens, I think they're opening light so I jam it in with a marginal or speculative hand)
punish the limpers a player or 2 or 3 limp and you jam preflop with a marginal or speculative hand...or even a strong hand for that matter.
small raise with intention of jamming almost any flop
Flat a small raise from the BB with intention of jamming any flop that connects with you, or doesn't connect with villain's perceived range.
For me, anytime a preflop raise would be more than about 25-30% of my stack I either make a commitment to jamming on almost any flop, or I just go all in preflop. This is to make my future decisions easier and maximize my fold equity.
notice how many times I used the words "intention" and "commitment" I think most importantly you want to have a plan for various stack sizes. That doesn't mean you 100% rigidly stick to your plan...but just having a plan is the first step....knowing when to adapt or even scrap your plan is a skill that comes with poker experience.