How to ask for rec letter for another program after verbal commitment to stay with my current advisor for PhD?
Absolutely you should apply to other programs, and your advisor should be encouraging you to do so.  If for no other reason, your advisor could be hit by a bus tomorrow, and then you'd be stuck without any back-up plans. 

Your advisor should have no qualms about you applying to additional programs. He should be willing to submit his LOR for you. Moreover, if you secure some interviews, it is such a unique opportunity to meet some other PIs in your field as well as students who are in your field. 

In sum, your professor should be totally open to doing this, just explain that you don't want to place all your eggs in one basket.  Good luck!