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.
I suspect that the answer to this question may depend on the situation where you are and your behaviour intentions after one and two doses.
I believe that the original clinical trials of the vaccines did not test different dose intervals. The idea that a longer gap might be helpful came, I believe, from other vaccines, where it's been established that a longer gap sometimes improves long-term protection.
I think a reasonable current estimate is that one dose provides somewhat less protection than two doses.
If you are in a place where there's lots of current infections, the short-term issue of getting more protection may dominate the decision, so pointing towards getting your second dose as soon as possible.
If you are somewhere with very low prevalence of the disease then it might be more worthwhile to try to optimise for longer-term protection, so could be worth digging into whether you believe that a longer gap is helpful.
You might also want to factor in your enjoyment of the next several weeks. If you're going to hold off on social contact whilst in the gap between doses, but will feel happy meeting people once you've had both doses, then it's reasonable for you to think how much extra enjoyment you'll get by pulling the socialisation time sooner.
Another factor that potentially pulls in the direction of earlier second doses is the likelihood of booster vaccinations. I suspect it's reasonably likely that we'll be getting boosters or vaccines for new variants each year, like with flu. That tends to make me think that the longer-term effectiveness issue is less important than short term, because you'll probably be getting more doses in the future that will help you with the long term.
I guess officially it's supposed to be 3 weeks, but IIRC, I heard a bit longer (~6-8 weeks?) is ideal?