+1 on putting a laser underneath the barrel. I have a dovetail mounted laser on my Marlin Model 60 and it's amazing how accurate they can be. Mine is zeroed at about 40 yards. It's as accurate as the gun is and more. The problem with red lasers of course is that they don't show up in the daylight. A green laser would be needed and even that might not be enough on a bright day.
I keep the laser on my Model 60 for use as nightime varmit control. They can be hard to keep zeroed in if you get a cheap version but even the cheap ones can be made to stay in position. It generally involved replacing the original screws on the adjusters. If you tighten them too much they will distort the laser though. With a little practice you can get them just tight enough however. Once in place you can drive tacks shooting from the hip. They're just right for night shooting of varmits like rats etc.. If you can get that dot on your target and pull the trigger you will hit your target. I've shot bats flying through the air from the hip with this setup. If you've ever shot bats you know that it's hard to do because they change direction so often. But with a well mounted laser you can do it easily.
I use the cheap lasers too. I got mine from a Bass Pro shop. I think they still sell the laser but it no longer has the barrel mount. They used to come with a barrel mount. Mine did. I did see a laser that claimed to be easily mounted to any gun on this web page
. I'm not sure how exactly they mount to anything besides a Weaver mount. Maybe they have barrel mounts since they say it can mount to almost any firearm, "with or without Rail Slots!"
There are mounts around for mounting to the bottom of a barrel. I've seen several in recent years.
I also ran across this setup
while I was looking for the links I had for laser and mounts. It could be exactly what you need.