I have large hands and thick fingers. My piano teachers, Cecil Lytle and (the late) Nathan Schwartz also have/had large hands. Cecil's make mine look small, yet he won an Int. Liszt Competition...the trick to playing between the black keys is to bend laterally (away from the thumbs) slightly at the wrist to fit your fingers deep into the white keys. Sounds awkward, works great. Nathan taught me this, he may have learned it from his teacher, Alfred Cortot.
P.S. I watched myself play last night, and I also bend medially, toward the thumbs, to fit. So, a loose wrist helps you fit fingers sideways. That was a movie, too, I think.