To post a bond (in US English) is to pay a sum of money to formally promise that someone accused of a crime and being kept in prison will appear for trial if released. It can be paid by the accused person, or on their behalf e.g. by a relative or friend, or by a person or company who provides a bond as a service for a fee.
Bond (Cambridge Dictionary)
'Bail out', a phrasal verb used intransitively, means 'to jump out of an aircraft with a parachute because the aircraft is going to have an accident'
Bail out (Cambridge Dictionary)
It is possible to speak of 'bailing someone (or oneself) out' of police custody. One would not, in any English variant, speak of 'bailing out' of custody.