If you are not going to carry supplies in your purse, carry a handkerchief.

The old way of dealing with this was a rolled up cloth between the legs. With a handkerchief you fold it around toilet paper and then play that between your legs held in place by your pants or shorts.

If you were wearing a skirt... My condolences,

(Not a girl but my knowledge of history helped out a girlfriend when were were out camping. And she had miss counted days.

Best part is you can wash and wring out a handkerchief easily and pat it nearly dry with paper towels or more toilet paper to use with new toilet paper shoved between it. Not as good on professionally made products but no one questions a bandana or handkerchief in a purse or pocket. Just make it apart of your EDC)