Not long after I got my 60D, I determined I wanted to go to full frame one day having come from 35mm slide photography 15 years previous. When I had the money, I replaced the 18-135 USM I had with a 24-105 f4L. The improvement in image quality was noticeable. However, I missed the extra 'width' and 'reach' of the 18-135, but I got accustomed to it. I still think the 18-135 is a fantastic all-around single lens solution on a crop sensor such as the nnD series of Canon cameras. Granted, the 24-105 f4L is not the level of image quality your 70-200 L can produce, but with micro focus adjustment on your camera, the results are very satisfactory.
When I needed a 'wider view' than what the 24 end of the lens could provide on the 60D (38mm equivalent), I ended up buying a 16-35 f2.8L to handle.
Some might think the 24-70 F2.8L or f4L would be a good end-to-end focal length 'match' to the 70-200, and it is. However, for a single-lens walk around, I value the ability to get the added reach of the 24-105 when needed over a very minimal loss of IQ. Whether shooting indoor events at church, railroads, cityscapes, or vacation photography, the 24-105 is on my camera 80% or more of the time. Also, my preference is overlapping zoom-ranges, so my 'magic trio' is 16-35, 24-105, and 80-200 (predecessor of your 70-200). That way, I don't have to change lenses when I need only a bit 'more zoom' one way or another.
Oh, and I finally did move up to full frame a couple years almost 3 years ago.