a 1" paddle bit will do the job. I've done it thousands of times without issue. about a dollar and maybe 3 seconds time. it aint rocket science folks.
Paddle/spade bits work great for drilling virgin holes in wood, but they are pretty useless for enlarging holes that are already close to the size of the larger bit (especially freehand) since they have nothing to pilot themselves in. You could force such in a drill press, but not a hand drill very well. But it could be entertaining to watch. In this instance the OP is trying to enlarge an existing 15/16" hole to 1".
Generally, the best bets for enlarging a hole already close to the right size would be twist bits (often simplest solution), hole saws, and Forstner bits, though the latter two will require something to pilot or guide them on initial entry (such as a predrilled block of wood to help constrain them while starting the new enlarged hole, after which they will guide themselves the remainder of the way and the guide block can be removed if desired).