Many men will wear hats for mincha and for maariv (also for other prayers if not wearing a talis). Many will not. And then I've seen a few who wear hats for mincha and maariv on Shabas but not on weekdays. This may, of course, be simply a way of dressing more nicely for Shabas (Orach Chayim 262), but I'm wondering (a) whether there's anything more to it than that, and, if not, then (b) whether there's any source that gives specifically this practice as an example of dressing nicely for Shabas.
Pri Tzadik Shushan Purim says that the Minhag by Klal Yisrael is to wear a hat on Shabbos that is different than the weekday, since the hat is indicating the crown of Torah which was received on Shabbos.
ועל זה נתפשט המנהג בישראל ללבוש בשבת על הראש כובע משונה בתוארה מאותה שלובשים בימי החול, והיא מקפת הראש כעין עטרה, מה שאינו כן בשאר הבגדים, שמחליפין אותם רק במה שלובשים בגדים חשובים יותר ולא בתמונה אחרת מיוחדת לשבת, מפני שהכובע ירמוז על כתר תורה שניתן לנו בשבת, מצד נשמע כנ"ל
Kal v'homer -- It is well held by the sages that on weekdays one should wear a proper hat when praying. Commenting on Shulchan Aruch 91:6, the Mishna Brurah holds that one should always wear a hat during prayer because it is not the custom to appear before prominent persons without a hat on. As discussed at http://www.theyeshivaworld.com/weekly_torah.php?id=19, there are differing views as to what to do if you have no hat, some holding that you should daven alone if you can't find a hat (and you can't use someone else's hat without permission). Assuming that that is the rationale for wearing a hat on weekdays during prayer, kal v'homer (how much more so) must it be true that a nicer, special hat should be worn for Shabbos, just as you would wear nicer clothes on Shabbos to be kavod haShabbos. Some Hasidim take this a little further and wear streimels (fur-brimmed hats) because in the 18th and 19th centuries in Eastern Europe, that was very impressive.