I space mine out for two reasons, first is to not agitate my immunocompromised and autoimmune system any more than it already is and the second is to better match the seasonal patterns that appear to follow holiday gatherings, so I use holidays as benchmarks.
Flu usually really gets going during the December Holiday Season, it crests around New Years and skids to a sort of quiet sometime after Easter so for this reason I have been aiming at just before Thanksgiving to get us through April.  We used to do it in October but everyone kept getting Flu type A in April and rolling the shot back a bit appears to have helped.
As for Covid, well this one seems to come on a bit earlier, probably because it is so much more contagious but at any rate it appears to sort of catch fire closer to Thanksgiving and then quiet down a bit around Valentines - St Paddy's. As a result, this one I do first and  as close to Halloween as we can get. 
This years Covid shot made everyone here sleepy so I'd go for a weekend where you can cozy up with movies if it happens.  I guess it changes every year just like the Flu shot.  Flu is next on deck. 
As an aside I also picked up a bunch of multi tests for Covid, Flu A & Flu B to have on hand to prevent anyone from needlessly visiting a medical facility during respiratory illness season and to save from missing work and all that, to me $10 a test is worth it. Last year we did the ihealth ones and this year we did the Binax but I saw different ones in stores like Wegmans and I htink Walgreens makes one too I think so it is good there are options now.  I would have loved this when my kids were little, no unnecessary missed days at school.