We’re going back to see the places that we loved so much on our walking tour, but just didn’t get enough of. This trip includes Ennis, Galway, Leenane and Westport. Is there a favorite place for music or a “do not miss” that comes to mind?
This doesn't really answer your question, but the best music we heard in Ireland were the buskers in Galway. Amazing.
Never been to Leenane, but
In Galway our favorite is The Crane Bar (upstairs). Tig Choili in "downtown" has good sessions but can get crowded and loud.
In Westport, the goto places are Matt Malloy's (founding player in Chieftains) and Porter House.
In Ennis, the sessions move around depending on the day of the week.
I find thesession.org very helpful. If you aren't looking for trad sessions, it (and I) are no help.
You might want to re-think the idea of just retracing your successful tour. There are lots of great places on the Westcoast that are as good as your mentions, not only for music, but for beauty and interesting things. Dingle, Doolin, Donegal, Clifden, Sligo to name a few. We've been to Ireland five times (12 weeks total), and never wanted for a great time and great music (although that takes some research).
We did another southwest sweep last May that included Kilkenny, Cork, Bantry, Allihies, Dingle, Doolin, and six towns along the lower Shannon River by boat. Trad sessions 17 out of 20 nights.