I have a quick change in the lid area of my lunch bag (it's a can cooler with a top area perfect for something small like that) Google for Ozark Trail Can Cooler, I got it at walmart for $7.
In the lid I keep a Tykables Tighty Whities for its utility, all white color, and good capacity/fit. I also keep a 1gal ziploc and a folded up black trash bag in case I need some disposal options.
In the front there's a zipper pouch, where I keep a travel size talc, zeesorb, and cornstarch. I also have a small plastic bottle with a few ounces of baby lotion in it.
The zippers on the lower cooler area and the upper lid came with pulls, but I removed the top pull to make it less obvious as an area for additional storage, and to prevent accidental opening by myself or someone else.