Partial answer as it deals with the idea of reading the Torah after all prayers have been completed. I am uncertain what occurs if you can open the ark and read after the amidah. Though, it seems that you could. After all, that's what we do during Shacharit.
I am excerpting this answer from another MY question. Kudos to @Yoni.
The mitzvah of reading from the Torah on Shabbos, YomTov and Mondays
and Thursdays takes place during tefillah but is not dependent upon
it. Certainly the preferable mode of reading from the Torah is during
davening but the mitzvah lasts all day and when a minyan is available
they should read from Torah. (Mishne Berurah 135:1, Shaar Hatziyun 3
and Aruch Hashulchan 135:6,7, Dagul Mervavah, Mahari Asad 58. However,
some maintain that if mincha time has arrived, the Torah should not be
read. See Mikraei Kodesh (Grodzinsky), Klal 13:5 and Shaarei Kedusha
I can't imagine that the rules of the fast day reading would differ from what was mentioned above. So, as long as it is still before sunset, it seems that you can still read the Torah by itself even if you have completed mincha.
I'm not sure what happens if you have past sunset, as it is the next "day", perhaps.