"Has any one have any experience hunting with these handgun? "
A little
I have to agree with Bill on his thoughts. While the .243 with a well placed bullet will do the job, it's my opinion it's getting a little marginal. Mainly because it's about 300 fps slower in the pistol than the rifle, limiting your range substantially, and with it's power modest at best to start with. The .260 isn't that much bigger, but enough to make the difference. And, as Bill said, the 7-08 is about as practical as one can get.
I do have a .243AI in that group, (top right,it's waiting for a new scope), I primarily use it for coyotes and Prairie Dogs, as it's also a slow twist.
I use to have a 30-40AI that was my "all-purpose", But as time went by and funds allowed, I set up barrels for game specific use. The last acquired is the other one waiting on a scopebase, is a .308 is going to have that "firefly" at the bottom on it and going Black Bear hunting this fall.
The one mounted on the frame, replaced the 30-40 as my deer barrel, which I just got recently and finished fireforming brass for today. It is chambered in the 6.5-257 Roberts AI. The one upside down to the middle right, is a .250 Sav. AIRLN. I set this up for anetlope which I haven't used it on yet, But it would work for deer as well. I used it last year for that purpose, loaded with 110gr. Accubonds. But it didn't get to draw blood as I passed on a small buck at 40yds. Which turned out to be the last chance I had to get out last year. The last one in the group, is a .218 Bee is my "shorter" range pest contol barrel. Along with those, I have a .300 JNKR (.300-376) being built for use on Elk.
All in all, the Encore makes for a pretty good hunting rig, but I still think you be better off with a little more than the .243..
Dave