Fender Ultra Tele Neck

I really like my American Ultra Tele but I do have a concern with the neck. It is the modern D neck. If I am playing "cowboy chords" on the first couple of frets, especially the C chord, I find there is not enough space and my fingers seem to mute the top E string unless I change my position. The rest of the neck is fine. So my question, is there another neck, perhaps not as thin that would help resolve this problem?

Hope this is not a stupid question and any help or suggestions you may have would be greatly appreciated.