So I've been traveling quite a bit recently, without good facilities for washing clothing.
This is relative, of course - I have much more than people throughout 99.99 percent of recorded history have had, which is plentiful hot water.
Anyway, been doing some laundry by hand and I've experimented with skipping ANY type of soap. Just hot water.
And what do you know - it usually works GREAT. No smells at all left in the clothing. For stains I may still use a little soap, but mostly I use no soap, and it makes it a lot easier, becuse there's less need to rinse.
I think I'll experiment with using zero or very little laundry detergent now. Mostly the smell is too strong for me anyway.
Another tip - after you're wrung the clothing out by hand, roll it up in a towel and step on the towel. It'll make the clothing as dry as after the spin cycle in a washing machine, and will line-dry much more quickly.
And my last tip - the type of cloth matters a great deal. Thick cotton will take forever to dry. Choose quick dry clothing, and then washing clothing while traveling won't be a problem at all.