Like Tinkster already said, after "M-x compile" you can change the command in the mini-buffer. Like: make -f /path/to/your/Makefile-custom
I've put the added line in my $HOME/.emacs file:
Code:
(define-key ctl-x-map "\M-m" 'compile)
This enables me (after restarting emacs) to use "C-x M-m" as a shortcut for "M-x compile".