Another method I find myself using quite often is like this:

-------------x-------------------------------------------------------------
-------------1-------------------------------------------------------------
-------------2-------------------------------------------------------------
-------------3-------------------------------------------------------------
-------------3-------------------------------------------------------------
-------------1-------------------------------------------------------------

... especially when changing from a C Major to an F chord...

You just play the bottom 5 strings and use your thumb on the sixth string.
You can also optionally leave the top E string open and get an FMaj7 to add a little flavor to a song. Again, once you learn this fingering you can move it anywhere up and down the fretboard.
If you can't use your thumb to properly hold down the sixth string... just use it to slightly mute the string instead and play only the 2nd, 3rd 4th and 5th strings.