Thanks to alert friends who know I travel a lot, I've encountered two useful products:
GoTubbs: From the same folks who make GoToobs, these are small pots with snap-on/squeeze-off covers. I use them for medicinal skin cream I need to carry around with me. They are not recommended for liquids, but the packaging doesn't carry the no-alcohol, no-oil, etc., restrictions you have on Go-Toobs. There are different sizes available. I use the small one (0.9 cu. in., 0.4 oz.), which is about the size of a lip-gloss container. It is very lightweight. Having ruined the lining of a purse via a leaking bottle of hand lotion, I put the GoTubb in a small ZipLock bag; I had no leaks over the course of a multi-month trip. These are available online, but mine came from The Container Store, which shelves them near the Nalgene bottles.
Silicon Bottle Handle: I can't find an online link for this (a gift), but I think I saw them hanging on a peg at The Container Store. If not, my guess is that it came from an outdoor-gear store. The brand name seems to be "Tag" (tagltd.com--product not found on website). It's a lightweight piece of black silicone shaped quite like a figure-8 except that the smaller lobe is a teardrop. That lobe is stretched over the top of a standard non-reusable water bottle and rests just under the cap. The round lobe might work on bottles with larger necks. The unused lobe can then be clipped to a ring on the outside of a purse, backpack, etc., with a connecting device like a carabiner. This is what I'll be using if I can't find a replacement purse with an exterior water-bottle pocket when my current purse wears out.