Reb Eliezer,
The Mishna Berurah is specifically talking about in his times where people did not go out in the street or meet with respectable people without a hat on. ובזמנינו צריך להשים בעת התפלה כובע בראשו כדרך שהולך ברחוב ולא בכובע הקטן שתחת הכובע כי אין דרך לעמוד כן לפני אנשים חשובים
Nowadays (and in the times of the Shulchan Aruch) it is not the norm to wear a hat in the street or when meeting with prominent people. Therefore, a hat is not required by davening. A kippah is sufficient.
If you are a person that is already yeshivish enough to have the custom to wear a hat in the street and when meeting choshuve people, then you have to wear it by davening too. But I doubt someone wearing a black hat in the street is asking whether they should also wear it by davening.