The Cowboy Repeater (also available early on) packs a bigger punch and is pretty darned accurate, but the iron sights on it are an absolute pain to use. Not good for situations where you need to keep track of multiple threats or threats that are moving in and dodging around. I use it for opening shots on slowly moving targets, then generally switch away to a 9mm or 10mm pistol for the close-in work. I guess it would be more useful if you don't have iron sights turned on.
The service rifle... has bad iron sights as well. Difficult to use and not as accurate as the varmint rifle. I put my SR away and switched back to the VR.
After that, you'll probably find the Hunting Rifle next. Or at least that's the next long gun that I ran across by level 8 as I got closer to Vegas.
My stealth/sneaky/sniper character won't ever be able to use the Sniper Rifle or the AMR, she doesn't have the strength for it. (Well, maybe the regular sniper rifle.) I suspect that the Varmint Rifle is going to be my favorite for a very long time because I love the iron sights on it.