I also have a 5DIII. The 24-105 is a very, very good lens. It is good at everything but it is the best at nothing. You can find a prime that will do any task better. Today I hauled around a bag of lenes and left this home. I ended up with some good shots, but kept changing lenses.95% of what I took was between 24 and 105. So I ended up cutting a photowalk short because I was tired and on top of that I got some dust on the focusing screen because I was constantly changing lenses in the wind. I should have just taken the 24-105L. Larger point. If you are going for a walk about, this is your lens. If you are going to a specific event, with a specific task, you can probably find something better. Public speaking? I'd go with the super sharp 135 f2L, but you will give up the IS. Keep your 24-105L. Leave it on your camera unless you have a specific task.
+1 But if I might add, the 24-105mm is the best single body solution there is for events. I use up to 3 bodies at some events by myself so I'm okay with my primes. But for one body, zoom range is very important. Why? Stuff happens spur of the moment at most events. If it's a birthday party or wedding of some sort... moments just happen, people who come to you wanting a group shot just happen, no one can run faster than zooming with a 24-105mm to catch the moment, and even if you somehow run faster than the fastest man in the world, it's just rude at an event.
So I definitely recommend the 24-105mm with an adjustable flash for pretty much any event. Especially if you're getting paid. They'll accept 100 mediocre photos of whatever needed to be taken over 5 top quality ones from a limited range, that's just how the business works. No one in the thousands of events I've taken have said wow those 5 photos are really really nice, I don't care that you missed the other 95 shots that we needed. For events, invest in utility first, and upgrade to higher quality gear from then on.