well, I found that in 2 or more color designs only long stretches of one color alone has produced problems, really.
If you interchange the colors often enough, things stretch more.
reason is that the carried yarn is like a "leash" that ties things together (that is why you must not carry it pulled too tight, either). The knitting will have much give as the carried yarns have.
now even the carried yarn gets used in a stitch here and there, of course. the stitch is a loop and therefore has some "slack" to give.
if you would do 1 MC 1 CC (main and contrast color) then the stetch would be limited but not gone.
if you do 7 MC an 1 CC, e.g. then that string of contrast color will hold things too tight.
sometimes a mix of methods can work well. e.g. you can carry yarn only on sections of your knitting and do the sides "mono-colored" - but that will not work in the round. (but is good for sweaters and the like or for gloves or socks that are not worked in the round).
you can also change a pattern to make "dots" of the other color in them, to give them more stretch.
if you work in little lines, e.g. you can just use an end of yarn seperately for all sections you need that strand in (therefore not running it around the whole sock) and then weave the end. That would work for vertical lines, diagonal lines and "local patterns".
this seems hard to describe, but I will explain further, if you need help there.
some patterns, though, will just never be practical for tube-knitting that gets worn as garment.
a size bigger on your socks, e.g. might end up being too lose and slide down...
another method is, by the way, to include ribbing into the tube, to act as a "spring". That ribbing you can easily work in 2 colors and use as a part of the design (if it fits and you like it) - e.g. k1 p1 with all knits in one and all purls in the other color. Result: extra stretch to the tube, just because of structure.
AND, of course you could do actual double knitting and have not stranded yarn at all, but that seems toooooo much for a project of socks (and the give is not so extended, either)
PS: where is that sock pattern or a pic of it? is it available online? then we could probably give more detailed tipps for that special occasion.