Hopefully someone will come up with some good ideas for you, this is way beyond my area of expertise. But I just wanted to let you know that my house does not have a coat closet either!! I wish it did. We set up hanging bars in the laundry room for our coats and when we have guests, we have a "coat couch"...less than ideal. ;)

What about something like these (as inspirations). Somehow I think the roof line of the vestibule ought to be different than a gable. Have an architect or design build type firm work with you for a suitable design.

Check your front setback requirements. Many houses are built right at the setback line and without a variance cannot be extended towards the street.

Also consider space -- where would a vestibule fit on your house. My first thought would be extending a gable to create a porch over the left side, and having the entry door face the drive with the closet facing you as you enter. Windows make it a tight fit.