And I'm not sure why you think inlining would result in the speed increase? I believe the compiler will inline it when the standard memcmp is used too.
Regarding inlining, using GCC 6.2 with "-O3 -march=native" on macOS I first had to remove "111111" and "222222" as constants, otherwise GCC precomputed both loops.
When I used __attribute__((noinline)) on mycmp, half of the difference went away. When I forced mycmp to check the alignment, mycmp became slower by almost a full second.
That was on a 2011 Mac Mini. Using a newer box with a Haswell chip (Xeon E3-1230 v3) and GCC 5.4.0 performance is about the same when preventing inlining and adding the alignment check, rather being slower. In both cases the assembly confirms that neither mycmp nor memcmp were inlined.