I used to own the 70-300 non-L, and I wouldn't recommend it as a replacement for the 55-250. It isn't an upgrade; what you are getting is a bit more reach (the equivalent of a couple of steps backwards), at the expense of the 15 mm on the wide end, which makes the lens much more versatile.
I now own both the 70-200 f4 IS and the 55-250. The 70-200 is a considerable upgrade in terms of focusing speed - which is why I bought it - and build. It also has better contrast and colour rendition, although these can both be corrected easily in post-processing for the 55-250. If you photograph a lot of moving subjects or sports, the 70-200 is a worthy upgrade from the 55-250. Having said that, the 55-250 gives tremendous value for money: I kept mine even after buying the 70-200, as my vacation/hiking lens, when I don't want to carry the heavier and more expensive 70-200. It never lets me down and has produced many of my favourite images over the years.
As for the wide angle: your current lens is already fairly wide (15 mm). While an UWA would give you more scope, you might find you don't actually use it that much.
I'd suggest a third option to consider: a fast prime or a macro lens. Your existing lenses are both fairly good, and a prime or macro would significantly increase your creative possibilities for outdoor photography. I absolutely love my 60 macro. It offers constant f2.8, plus close-focusing capability and incredible sharpness. It does a terrific job on portraits, landscapes, close-ups, long exposures ... you name it, this lens can do it. It's tailor-made for crop cameras, and a lot of fun to use. However, there are a lot of other very nice prime lenses out there. Just a suggestion!