Obvious question: Where do you plan to hike?
Answers range anywhere from late September to late November, depending on what part of the country you're visiting.
Here in MO, we're just at or maybe barely past the peak now, the last week of October. However, it has been a mild fall and we're running a week or so later than usual.
In New England (VT, NH, etc.) the leaf-peepers head out in early October.
Farther south, later...naturally...
Anywhere at significant altitude (say above 4,000'), the season will vary with altitude as well as latitude. Keep in mind, though, that colors stop about 8,000' or so in the Rockies and Sierra because most everything uphill from there is coniferous not deciduous (except for scrub and brush, some of which can be colorful, too). The really colorful stuff out west, mainly aspens, grows in a fairly narrow band of elevations.