Orrrrr...
Just pick up a lightly used 28-135mm. So many have been sold in kit with various Canon cameras, that they are widely available for $250 or less, often practically unused.
Image quality is just about the same as the 24-105mm... Especially at the focal lengths they share. The 24-105mm has stronger vignetting at the wide end. The 28-135mm is slightly softer racked all the way out at 135mm (but improves if stopped down). Focus performance, close focusing ability, and IS effectiveness are all about the same, too. The L is better built, as you might expect, and possibly better sealed.
But if you only spend $250 or less on a walk-around lens, maybe you'll have enough budget remaining to also pick up another 17-40mm, so you'll be covered all the way from UWA to moderate telephoto.