My son just started putting on tefillin to practice for his upcoming bar mitzvah. He wears a hat and jacket while davening shacharis but does he need to wear a hat and jacket for mincha/Maariv and on Shabbos when he doesn’t wear tefillin? We don’t have an established family minhag.
There are different customs in different communities. If you don’t have a specific minhag, let him do whatever the local minhag is, of what the boys in his school are doing.