I tried using the small hole but the resistance to letting the probe through was pretty high, so I chickened out. My Maverick probe is an older one, it might be a bit thicker.
Many years ago I bought a professional tire repair kit that allows me to fix tire punctures just like they would at a garage. The kit included a bunch of what amount to heavy strings coated in rubber, a radial wire separator, a patch insertion tool, and a small jar of lube. You're supposed to thread one of the rubberized strings through the eyelet at the business end of the insertion tool, dip the tip in the lube, and then push it into the hole in the tire. As you remove the tool the string drags with enough force from the tire that it pulls out of the tool and stays in the hole. You re-inflate the tire and drive away. The heat from friction on the tire vulcanizes the rubber in the patch string and fuses it with the rubber in the tire. The excess hanging out of the hole wears off.
Why did I mention this? One time I forgot to use the lube on the tip of the patch insertion tool. I pushed and twisted and shoved for all I was worth and I could not get that tool to go through the hole. I must have worked at it for ten minutes and got absolutely nowhere. Then I noticed the jar of lube. Facepalm! I dipped the tip of the tool in the jar, picking up just a tiny bit of the lube, and gave it one more try. With almost no effort at all the tool slipped right into the hole. Innuendo aside, the point here is a little bit of lube applied in the right place can really make things a lot easier. I suggest a quick wipe with a paper towel containing a few drops of vegetable oil. I suspect you'll find the probe vastly easier to get into that small hole.