I also have the T3i.
I went from the original 18-55 (from my 300D) to the 15-85. Shortly after that, I also bought a 50 1.8.
As noted, you need to look at your needs.
Personally, I love the 15-85 for its range and speed/accuracy of auto focus. Depending on what and how you shoot, the "you already have the 55-85 range covered" argument may be a bit short sighted. I also have the 75-300, so a similar argument is you won't miss the 55-75 range, so your covered. However, for me, the 55 was never quite long enough and I wasn't sure 85 was going to cover it. I really wanted the 18-135 for the range, but we have that lens at work (not the STM version) and I really don't care for it. It just never seems quite sharp enough and it just doesn't have a great quality feel to it (this is perception - not necessarily a practical reality). Having used the lens for about 9 months now, I am pleased with the range for my type of shooting (but maybe not for yours).
The other consideration is that since getting the 50 1.8, my biggest disappointment with the 15-85 is that it isn't a fast lens. Take others advice and pick up a fast prime - it really does add a whole new dimension to your arsenal of tools. I say this because (and this again depends on what you shoot), with slow glass in most lower light settings your going to be as wide open as the lens will go because the other option will be a higher ISO or a slower shutter speed (and you are usually pretty slow to begin with in lower light settings) meaning that adjusting the aperture really isn't an option because you are already maxed out so to speak. I find that most of my shots are wide open. With the 1.8 I now have some flexibility that I didn't have before. The 18-55 lens will be around 5.0 or 5.6 at 50 mm vs 1.8 with this prime. I think that works out to somewhere around 3 stops. With this option I now find myself stopping down for a variety of reasons depending on the situation, where I wouldn't have if I was stuck at 5.0 or 5.6. My favorite feature of this is the OOF background it can produce and because of that alone, I now love this rather cheap lens and really want more fast primes. I can be an addiction.
Having said that, it is not easy shooting a narrow depth of field and one reason why the 50 1.8 is so great. You can practice with one for around $150 and probably get half back later if you decide to upgrade to the 1.4 or better.
For me, what I've come to realize is the 15-85 is a great lens for most of my general photography and the prime fills in where the fast aperture can be a benefit (usually portraits). I plan to add a few others (feel free anyone for suggestions - thinking about the 85 1.8 for maybe the Sigma 30 1.4 - trying to figure out what I'll use more) as I think this is the best fit for my needs.
What I found was, on a crop f2.8 isn't that fast in terms of OOF backgrounds and with the same basic range of the 18-55 I don't think I would have been happy with the 17-55. I prefer the 15-85 with a few primes.
Using the 15-85 with the 430EXII is an exceptional package and if you use a flash frequently then an external flash should be another upgrade path to consider. This combination produce great results for family pictures indoors because the range is ideal, it focuses fast, the flash recycles much faster than the built in and with the ability to bounce off the ceiling, typically more pleasing lighting (this is another art in itself).
One argument I have seen elsewhere, but not here, is to buy the lens for what you are shooting now. FF lenses don't have an optimized range for crop. Again it depends on your needs, but for me 24 isn't wide enough (24-70 lens). The thought here is good glass holds it's value and if you go FF later you can recoup most of it and benefit from the optimized range in the mean time. Many here think of whatever you do lose as a rental fee of sorts.
And I planned on keeping this response short.