Bash (or any other shell, really) is not a good tool for general programming. Bash doesn't even handle floating point arithmetic, let alone anything more complex. The shell is a shell, you can write simple little scripts in one, but you shouldn't think of it as a general purpose programming language: it isn't.
I'm afraid that if you insist on using bash for something like this, you will be forced to use more and more complex workarounds, starting with printf:
$ printf '%x\n' $"$((0xa+1))"
b
The good news is that you can at least also do this:
$ printf '%x\n' 11
b
So you can easily convert from one to the other, which means you don't need to call printf every time:
var=0xa
let $((var++))
echo "Var (decimal): $var"
printf 'Var (hex): %x\n' "$var"
Running the above would print:
$ foo.sh
Var (decimal): 11
Var (hex): b
In other words, you can do your thing in hex and not care about how it is displayed until you need to print something out.
But no, you won't find a native way of doing this sort of thing in the shell because you are trying to use the shell for a purpose it was not designed for.
