50 miles west of Clovis on U.S. 84/60, you'll pass through Fort Sumner on the way to Albuquerque. It was all we could do not to take a short detour when we saw the sign for the gravesite of the legendary Billy the Kid, a 19th century old west outlaw. The Kid's gravesite sits within an iron cage behind the Fort Sumner Museum. There is an admission charge for the musuem but visitors can go around to the small graveyard in the back for free. The iron cage is to thwart would be headstone robbers. The gravesite is a good bit of American history.
Look for the signs for the gravesite and/or for the Fort Sumner Museum. You'll take a road (can't recall the road number) for about 4 miles to the south of U.S. 60/84 to get to the gravesite.
Visit the musuem to get more details on Billy the Kid's showdown with Sheriff Pat Garrett and about the Old Fort itself and its relationship with the Apache and Navajo Indians. Admission is $3/$2 children. Visitors can go to the back behind the musuem for free to view the gravesite of Billy the Kid.
If you must go off the beaten path the thing I would recommend would be the dry salt lake South of Clovis, but East of Portales. It's near Arch, NM closest to the Texas border. It's a big dried up salt lake. If you find a road out to the lake, you can dig in the dirt and find some really great salt crystals. I know it's a bit of a stretch, but hey, your trying to find things to do in Clovis, NM.
Another place to check out would be the Oasis State Park. It's south of Clovis near Cannon Airforce Base, it's a 3 acre lake which they stock with trout in the winter and catfish in the summer. The best thing to do there is bird watch. They have a lot of great trails and you can run into bobwhite, american kestrals, and blue jays. It really depends on the season.