I have a hat pattern knitted on 10mm circular needles. When I knit my tension square, I come up 1 stitch too many. I generally need a half size larger than the one stated on the pattern for most things but I can't find an 11mm circular needle.
Should I knit a size smaller, with the thinking that if I'm gaining 1 stitch per 10cms over 50+ cms, it will come out the size want?
If you end up with 1 stitch too many, then your hat will come out smaller than you want it to. You're not actually gaining a stitch, you're knitting too tightly.
For example, if you are to cast on 50sts which should measure 50cm, that means your tension should be 10sts to every 10cm. If you knit a 10cm square and end up with 11 sts in it, then your 50sts cast on will only measure about 47cm in total.
If you can't find a larger needle, can the pattern be adapted so that you can add a few more stitches, if it's just a simple stocking stitch hat? Or you could knit a larger size on smaller needles. Try out different needles and knit another tension square until you get the tension you want. The other thing you could do is just knit it with the 10mm needles and stretch it by blocking after you're finished, if the yarn is suitable to do this with. Just make sure your cast on edge is loose.