There are plenty of solid bang for the buck A orAA EDC's around. Give a price point so we can further direct you as to which lights to look at.
Under $20 sounds about right.
Check out the Fenix E01. Puts out ten lumens , which is sufficient for quite a few tasks. Buy one, see if it works for you and go from there. Also, do not use alkaline batteries for long term storage. They have a high chance of leaking and ruining your light. Lithiums would be much better.