Is there a jurisprudential school of thought that permits a woman to cover only her mouth when going out to the market, without covering the rest of her face?

There is no recognized school of thought that mandates covering the mouth without the rest of the face. Rather, the preponderant opinion is the obligation to cover the entire face except for the eyes, and there is also an opinion that does not obligate covering the face at all. What women do in this regard is perhaps a matter of custom.