Jill, I never remove mine just add fresh when it's needed anytime of the year. The old mulch will break down and enrich the soil. Others may not agree, but that's the way I do it. :)
Jan

Jill, I agree with Jan. I never remove old mulch. I just put the new mulch on top of the old. But I don't consider old daylily leaves "mulch". I remove all old DL leaves for rust prevention. I like to add new mulch (pine straw and shredded wood chips) in the spring, right before everything starts to flower. It makes everything look new and fresh. However, I also add it whenever I can get it for free - any time of the year. I do try to keep the mulch pushed back away from the base of all my plants - this helps to prevent rot.
Kathy