The FOM spells it out. Coat has button and stripes. Jacket is "leather".

This is EXACTLY from the FOM that everyone keeps quoting.

Coat:
• Must be appropriately buttoned in
public view


Leather Jacket:
• Black leather jacket with approved
logos embroidered on lining (if
desired)
• May be worn for operations within
the domestic U.S. and for operations
between the continental U.S. and
Hawaii, Alaska, Canada, Mexico,
and the Caribbean

Hat:
• With United-issued hat emblem
• Uniform hat must be worn whenever
the uniform coat is worn


So the "hat" is only required with the "coat", which is not the leather jacket. So no coat = no hat, since leather jacket "may be worn".

Quod erat demonstrandum.