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A couple of thoughts.
First, this indicates you are a person of at least modest wealth, or at least quite comfortable. The vast bulk of Americans (farmers, workers) could not afford such an excursion.
So there's a good chance you bought this at a specialty store, like Abercrombie & Fitch (back when it was a gentleman's outfitter store, not a hip kiddie's mart). So you bought what ever company they had contracted with to put their own name on knives.
You already are carrying a fixed blade, which will get used for skinning and larger chores. Since you're with a group, the guide probably has an axe or at least a hatchet.
So what would you use this small knife for? Small chores, possibly some food prep (although again, the guide probably did that.
I vote for a folding knife with the following:
--a main, general purpose blade
--an awl. Horses and leather will be everything on this trip, so it would be good to have something that at least makes extra holes.
--can opener. For rations. Canned food was big back then.
--possibly a bottle opener. The crimped bottle cap had already been invented.
--a corkscrew. You're a gentleman, right? Thus, there has to be drinking. Keep in mind that hard liquor came with corks back then, and not just wine.
If it's after the Great War, things get easier, and the class lines diminish. I just got through reading Bernard Levine's book, Pocket Knives. Remington started in 1920 and quickly became a giant. At its peak it produced 10,000 "high grade pocket knives" a day, with "over 1,000 pocket knife patterns, along with hunting knives and household cutlery." Winchester started making knives around the same time, and was very big in hardware stores. So if you were a gentleman or a farm hand or the guide, there was a reasonable chance you had one of these knives.
Mmmh! Interesting challenge. Fine idea.
I think our man in this scenario would be well-heeled and had travelled to Europe, or was a wealthy immigrant from that continent to the USA. As pointed out, the trip although authentic smacks of well-off authentic and most of the tools/equipment would be covered. Or a servant would be tasked in this fashion....
Assuming this, our man would have a fine Sheffield knife in his pocket for personal cutting tasks. Single spring Equal End with Spear/Pen or Norfolk with Wharncliffe/Pen from Rodgers et al shod with epic STAG from now alas long extinct herds. Gent needs to cut plugs of tobacco, cigars, tit-bits of food even post might be delivered etc you never know....