It depends on the location, the species, the licensing requirements, the outfitter/guide’s availability, and — potentially — the cost of the hunt.
Most commonly, for hunts with over-the-counter tags available to non-residents (which, sadly, are becoming fewer and fewer these days), booking in the early spring (February or March) for a fall hunt used to be standard operating procedure. That’s why late winter/early spring is the time during which most sport shows were/are held.
However, a great deal of this is changing for many reasons. With quality, highly-rated outfitters like you’ll find on OutfittersRating.com, their hunts are frequently in high demand. This may require you booking a year, two, or even three in advance! Sometimes in drawing situations, outfitters will take your booking based on the first season upcoming in which you are successful in drawing a tag. In this kind of deal, basically your booking has secured a spot in the camp, then once you draw you and the outfitter need to figure out the details of dates that work for both of you, etc.
In many cases, outfitters guiding on any kind of public lands will be limited to the number of clients they can host in a season and/or the number of animals that can be taken from their area in a specific season. This situation, too, can require booking a season or two in advance.
With the price tag on distant, high-adventure hunts in particular, booking several years in advance allows you time to save money for the trip as well. Making the booking and putting down a deposit with a quality outfitter provides real motivation to save the required amount of money for the trip, too.
Best of all, booking far in advance allows you plenty of time to savor the great anticipation of the adventure to come — so you’re getting the most pleasure possible from the money you’ll spend.