I guess officially it's supposed to be 3 weeks, but IIRC, I heard a bit longer (~6-8 weeks?) is ideal?
TL;DR. It probably doesn’t matter too much, but there’s little risk and potentially a small upside in terms of total immunity in waiting up to 12-16 weeks for the second dose, at the cost of lower immunity while you wait. Prob not worth worrying about.
The period of 3 weeks between vaccine shots was chosen to minimise the time to complete the trials, not to maximise the efficacy of the vaccine.
To a certain extent I would say that it doesn’t matter when you get the second shot, as you’ve already done the most important part in getting the first shot. You’re no longer going to die of Covid. However I guess that you are trying to maximise both the efficacy of the vaccination and the duration of the protection.
Here it’s difficult to have a clear view because the situation is evolving rapidly and you also have to weigh the advantage of increasing your protection from (say) 80 -> 94% rapidly against the potential added benefits of waiting 3 months. And this with the continued uncertainty of the variants and knowing that you will probably need another shot within the next 6-12 months.
I can’t currently find research for Pfizer but for AZ research shows that a gap of 12 weeks between doses leads to higher protection https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3777268
This is likely also true of all vaccines. The trade-off is the lower protection while you wait.
The vaccine is sufficiently effective 3 weeks later that you should get it 3 weeks later and resume your life as quickly as possible, at least in USA where there's enough to go around.
It is possible that 6 weeks is slightly better, but there's no way you'd give up those 3 weeks in between to get that effect even if it's true.