Not entirely:
"Research from Texas recently showed that mature bucks only used 30 percent of their home range during the rut, had two or more points of activity that they focused on, and they re-visited these locations roughly every 20 to 28 hours. What’s more, these same sites were targeted by numerous other collared bucks, possibly supporting the theory that bucks space their visits to doe groups to continually assess female receptiveness."
https://www.deerassociation.com/10-things-know-mature-buck-movements/
But sure, deer travel, and not just during the rut:
"Deer take short vacations. Today’s increasingly accurate GPS collars have taught us that deer (does and bucks) go on
excursions, or brief trips outside their established home range. They do it regularly and even on well-managed properties with high-quality habitat. Excursions have been documented across various landscapes, in all age classes, and just about year-round; however, there is certainly a huge spike in this type of movement during the rut. Unfortunately, there is no way to predict when and how far a buck will go."