Which communities wear tefilin while davening Minha on ordinary weekdays? I was told by my Rav that some do.
Technically the Zilberman community in the old city do, but they wear them all day anyway
The Teshuvos VeHanhagos mentions a few:
- Some communities in Israel (now and during the time of the Reishis Chochma)
- R' Velvel Brisker
- The Rama MiPano
Regarding your seeing people wearing tefillin during mincha, there are several possible explanations:
1] They may have missed shacharit or wearing Rashi tefillin for whatever reason, so they endeavor to fulfill that day’s mitzvah of tefillin at least during the mincha prayer rather than at some other time of the day.
2] Since technically speaking the mitzvah of tefillin is to wear them all day long, even though this is not done, some choose to wear Rashi tefillin at least during mincha in addition to shacharit.
3] According to the kabbalists, in addition to using Rashi and Rabbeinu Tam tefillin in shacharit, one should use Shimusha Raba tefillin in mincha.