an all - around lens is probably going to be a compromise. from what I have read, all of the 18-200+ lenses are a major compromise. but if you only want 1 lens - and need all the range - that's all there is. and they're all softer than their smaller zoom range counterparts.
I like a 2 lens solution myself - I carry the 15-85 and the 55-250. The 55-250 is pretty light and small - easy to take along when I want more range. That being said, I love my 15-85. takes great shots, my copy is sharp and has great contrast - and has a pretty comfortable range for me. I guess it could be considered one of the longer "standard" range zooms out there.
but it's another compromise as well. there's some distortion in the wide end - and it does soften images somewhat to straighten them out with DPP. for the most part, the distortion isn't an issue for what I like to take pics of.
I can't speak for how you like to take your pictures, but if there was the space/weight available for your trip. I'd add the 10-22 to my bag. but then again - for me, getting out to the grand canyon would likely be a once in a lifetime shot - and I'd compromise other places in lieu of a low-distortion wide angle lens.
Depending on your trip - and if you didn't need a huge telophoto - a fairly cheap/compact option might be a 10-22 and a 28-135. that'd cover lots of range, and be on the cheap. The 28-135 is a well proven unit. The primary basis here is going to suit you best in sunlit conditions. Then again I can't imagine visiting the canyon at night. although sunset and sunrise might be rather dramatic.
lensrentals.com is a pretty reputable site.