Suggested lens for Nikon D3100 for Lower Antelope Canyon, Zion, Page, GCNP?

My wife has a D3100 with its standard lens. Would photos be substantially better with a wide angle. Any suggested lenses or ranges? We won't buy more than one additional lens, but I'm not sure if it's even important to have another lens.

Just about any landscape photos can be enhanced using a wider angle lens than your 18-55 mm.

I used a 12-24 mm when shooting the slot canyons in northern Arizona a few years ago.

I used the same lens when shooting throughout southern Utah, from Moab to St. George

Note: There's a Nikkor 10-22 mm lens that is a little less expensive and Tokina makes a 12-24 mm that may fit your budget better. Tokina makes some of the best third part lenses for Nikon and Canon cameras.

If it's just for one vacation, why not try renting a wide-angle lens? That way you can try it for a few days, and if you really like it you can consider buying one later.

If you're looking for super wide-angle (your 18-55mm is already considered "wide angle" when you zoom it all the way out), check out any of the following:

Nikon 14-24mm (one of the sharpest wide angle zooms there's)
Tamron 12-24mm
Tokina 12-24mm

You could just head to the nearest camera store and ask them if they have any super wide-angle lenses for rent. They'll help you out.

Good luck, have fun in southern Utah! I love going down there!