I've noticed an interesting thing with my playing over the last few years. Perhaps the most creative period of time when I practice is the first ten minutes or so. Some pretty astounding little musical ideas have just jumped out of my guitar during that time period, things that lead me to either learning a new tune or adapting and (hopefully!) improving tunes I already know. But if I'm in a lazy kind of mood my fingers just seem to want to revert to the same old, same old right off the bat. Then what seems to happen is I quickly grow tired of what I'm playing and try to force the creative process, which of course NEVER works! So instead of playing for a long time I end up putting the guitar down after a few minutes.
So what I'm getting to is this. Maybe it's a good idea to take a minute or so before you start playing and just hold your guitar and think about things. They really don't even have to be musical things. I once came up with the changes to a song long buried in my memory by doing just that. It came to me as I was reflecting on playing with a friend long ago and the fun we had and the songs we played.
This is necessarily a private time or moment; if you're in a performance situation or sitting down to play with others you don't have the luxury of being introspective for very long. Enjoy those private moments!
There are other factors that impact that crucial first few minutes of playing, things like your overall mood and how energetic you are. If you've been doing something with your hands that is physically taxing it might be wise to wait a while before you play because the results will most likely be unsatisfactory.
What we all strive for is having the playing experience be rewarding and fun. Why the heck do it otherwise? So if that "first song" you always seem to play is one you really like, by all means, don't abandon it. Just think about maybe pushing it back for a while. You may find something pretty remarkable comes out in those first few minutes.
Peace & good music,