Again, depends on animal size. I am working with large framed angus with average mature weights around 1300-1400. I originally built my working chute based on some old university specs many years ago @ 28" wide - tore it down within 6 months. Like Dun says, an inch or two doesn't sound like much, but it's a world of difference if a cow feels squeezed or comfortable walking down the chute.
Remade the handling/breeding chute such that it was 28" wide up to ~2' high & then opened up to 31". Then I made my dirt/sod 12' long cattle truck loading chute 32" wide to insure a smooth flow of cattle onto the truck. The only issue at that width is occasionally a couple of 5-6 wt calves will try to enter as a pair - but one usually wins & both make it up the chute just fine. I will say that a dirt/sod base seems to be much easier to move cattle up than the wooden floor design as it is "natural" for them to walk on sod - not so for wood. It took a little more work to fill it in with dirt & plant sod but I have no regrets - advice from an old cow man that was in the business longer than I was old at the time