Overlap, despite what you hear on POTN, can be useful.
The 55-250 is an excellent value for the money- you will not find any better lens for that price range.
Think of overlap as a value- if you had a 18-55/55-250 combo- if you had the 18-55 on- and you needed more telephoto- your only recourse is to swap lenses- which might mean missing the shot - or more constant lens swapping. An 18-135/55-250 combo means that often, you don't need to swap lenses as much. the 18-135 is also a much better lens for general walkabout, I think, than the 18-55. As a single lens by itself- it's a good choice for going out and about with only one lens on the camera.
I find the 18-55 suitable for relatively close range- like indoors- but- it's still fairly slow- and limited range.. The 18-135, on the other hand, handles indoors as well as the 18-55, but it's beauty is outside- it's more versatile- and often negates the need to swap the lenses out. Someday- take a look at the exif info on your pics and determine what ranges you are using.. Most of the time I had a 18-55, I found I was using the extremes of the two- 18mm and 55mm- which means I am butting the ends of the range. To me, that means I probably could use extra range out of the lens. To me, that would be either a 15-85, 24-105, or 18-135...
The new 18-135 STM is a completely different animal from the original 18-135 IS - btw.
EOS 5D III, EOS 7D,EOS Rebel T4i, Canon 70-200 f/2.8 IS II, Canon 24-105L, Canon 18-135 IS STM, 1.4x TC III, 2.0x TC III, Σ 50mm f/1.4, Σ 17-50 OS, Σ 70-200 OS, Σ 50-500 OS, Σ 1.4x TC, Σ 2.0x TC, 580EXII(3), Canon SX-40, Canon S100
Fond memories: Rebel T1i, Canon 18-55 IS, Canon 55-250 IS, 18-135 IS (Given to a good home)...