H.L.
I can relate. I am VERY sensitive when it comes to any kind of odor on my clothes. What happens for me is my clothes get mildewy smelling because I am lazy and I leave them in the washing machine too long. You know, I throw a load in and then forget about it until the next day!!
Anyway, what I have found that works (although I like the tablespoon of bleach idea and am going to try it) is to put a 1/2 cup of table (white) vinegar and 1/2 cup of baking powder along with your normal detergent or whatever. This actually works very well. I had a load recently that I thought I had ruined because I just kept on forgetting it!! I think I re-washed that load 3 times!! One one time I put just the vinegar becauase I didn't have any baking powder in the laundry room (see how lazy I am, it was just upstairs in the pantry!!) and that didn't work, it's the combination of the 2 that works.
LIke others said though, if you want to avoid it, best is to rinse the item out and then hang or lay it out to dry before putting into the hamper.
Good luck