Change default compiler in make

In my Ubuntu machine, I have a makefile, and I want to compile the project to a different compiler then the default gcc compiler.

I read a lot about it, and I tried different methods, including:

1. Adding CC=<other_compiler> in the command line: make CC=<other_compiler>

2. Adding CC and CXX as an environment variable in the command line from which I execute make.

3. Adding CC=<other_compiler> inside makefile

But all of them didn't change the fact that the default compiler (gcc) was called, instead of mine compiler.

Any ideas what could have I been missing, and where else can I check?