I found this inexpensive tool for clearing snow blower chutes on Amazon. It was about $10 (search for "chute tool"). The tool is sturdy and entirely plastic so it won't scratch the paint. It came with a holder designed for the handle on a push snow blower, but it was easily modifiable for my tractor. I had to cut off 4 mounting nubs with a hacksaw and re-drill the mounting holes to accommodate an 8-32 screw. I used 4 of these (3/4" long), 4 lock nuts, and 4 small fender washers to mount the tool on the chute hydraulics cover. I peeked into the hole on the hydraulics cover where the hoses go in to confirm that my added hardware is nowhere near the mechanisms under the cover.
I decided to mount it on the snow blower because I didn't want to have to walk back to the garage if there was a jam. I thought about mounting on the steel upper deck of the snow blower, but if I decide I don't like it or it breaks, I don't want the holes in the steel. By mounting it on the hydraulics cover, worst case I have to replace the plastic cover if I make a change in the future. The black tool blends into the cover pretty well.