To stay in an unsatisfactory job, relationship, or situation while actively looking for a better one; literally, 'to ride a horse to find a horse'.

horseman; rider; mounted soldier

martial arts stance resembling the posture of someone riding a horse, characterized by a half squat with wide legs

to search for something that is already in one's possession or right under one's nose; to seek a better opportunity while holding onto a current one, such as job hunting while still employed