You’re right: since calling strcmp() adds up the stack management and the memory jump to the actual strcmp instructions, you’ll gain a few instructions by just checking the first byte of your string.
For your curiosity, you can check the strcmp() code here: http://sourceware.org/git/?p=glibc.git;a=blob;f=string/strcmp.c;h=bd53c05c6e21130b091bd75c3fc93872dd71fe4b;hb=HEAD
(I thought the code would be filled with #ifdef and obscure __GNUSOMETHING, but it’s actually rather simple!)