Joseph & DaasYochid:- Let’s get this straight once & for all:- Nothing ever gets in the way of commencing Tefillo b’Tzibur;
Certainly, not lack of a hat, and certainly pray Mincho before sunset without a hat, rather than after sundown with a hat; Certainly never ever arrive late to Shacharis, but always come at least 5 minutes early, so that you have Tallis & Tefillin on before Shacharis commences, so can get up to Tefillo b’Tzibur without skipping:- Only thing that can possibly be skipped at Shacharis is a hat by dint of your Tallis on your head.
If running to catch a Minjan is going to put you out of breath and jeopardize your health or state of mind, forgo the hat to have time to walk biMenuchas Nefesh, but don’t ever risk missing Tefillo b’Tzibur for any lame duck reason such as a hat.