Will horses (and riders) do things they wouldn’t normally do at home when they are out hunting? YES!! Does that build confidence? Maybe. Sometimes, after the fact, I think both human and equine might look back on an instance and think THAT WAS WAAAYY OUTSIDE of my abilities! THANK GOD IT WORKED!!!
I would think of jumping while hunting similar to “schooling” jumps and “showing” jumps. You might school 3’, but only show at 2’9". You might get away with jumping something unusual at 2’9" if he knows to keep up with the herd, but if it’s too unusual or well outside of his comfort zone jumping you might go over the jump…just without your horse and have ruined his confidence in YOU!
FIRST be sure that he is comfortable in body from the bottom up. Be sure that your saddle and bridle/bit fit him WELL.
SECOND be sure that you are riding him well to the jumps. If he has bruised your confidence, then you might not be able to support his confidence. If so, only jump things that he will jump with his eyes closed ( and yours too). Build from this. ground poles, 18" cross rails, 18" vertical, 2’ cross rails, 2’ vertical. Gymnastic lines of the height he is willing to jump with his eyes closed.
THIRD introduce him to hunting WITHOUT jumping. It is ALOT to absorb. Especially if horse and rider combo are not familiar with hunting.
FORTH take him out hunting with someone in front that knows if its a jump that is within your limits. For example, Saturday before last out hunting we came upon a coop. I was middle of the field. I noticed that the center of the coop on the landing side was a bit boggy. On the second run, I let the person behind me know, if possible to keep her horse to the right of the coop to avoid the boggy take off. As another example, being the visitor at a joint meet, the person in front of me let me know the upcoming coop had a drop on the landing side.
FIFTH once you have done the above you can PUSH your horse a bit more. Example. Thanksgiving meet. We are on a FULL CRY RUN on a red fox. The fox goes into a field we don’t usually enter. The coop has a rider that has been nailed in, plus a second rider that has been nailed above it. Two masters jump the coop + riderS. I’m next. I bring my horse from a canter to an extended trot, jump the coop and keep going. Master number 2 later in the day " Why did you trot the coop?" Me “So I had room to add leg!” The coop + riders was at the top of a 4 rail fence, so we jumped pretty darn close to 4’6" that day. I was glad for the herd mentality, but also glad to have a horse that would listen (came back to a trot) to make it happen. Lots of seasoned riders and horses had stops at that coop that day.