In ultimately omitted dialogue from the final draft script of "Anomaly", it was established (in information Reed told CommanderTucker) that Fuller had gone through tactical training with Reed, that Fuller had a daughter who Reed had met a few years prior to 2153, and that Fuller had been "hand-picked" by Reed to serve on Enterprise. The script also established that Fuller's daughter had been in either second or third grade when she and Reed met, and that she had written a letter to Reed, after the ship left spacedock, in which she had asked him to keep her father away from dangerous missions.
There was also a female Crewman Fuller aboard Enterprise. In "The Expanse", T'Pol mentioned that "Crewman Fuller" informed her that a shuttle was arriving at Enterprise with two hundred snow beetles. However, since T'Pol did not use a specific pronoun to refer to this Fuller, it is unknown to which crewman she was referring.