On the hunt for him VS On his hunt

No, on his hunt would not readily be understood by most English speakers.

It is risky to try and explain why particular expressions are not idiomatic, but in this case the fact that hunt takes "for" rather than "of" in this sense.

In fact, there are two related but different senses of hunt, which take different arguments:

Transitive hunt means "pursue a person or animal, to capture or kill them/it", as in hunt the stag/bear/fugitive. When it is used as a noun, it can be specified by a noun modifier, but not normally by a possessive or prepositional phrase: a bear hunt, but not *the bear's hunt or *a hunt of the bear. A hunt for a bear is possible, but shades over into the other meaning (below). To get a nominal phrase with a definite object we would usually say hunting the bear.

The other meaning of hunt is search for, and that takes an indirect object with for (not of). So hunt for my glasses means "search, trying to find my glasses". Hunt for the bear would normally mean "try to find the bear", while hunt the bear means "chase or stalk the bear, to capture or kill it".

This sense can be nominalised as the hunt for the bear but not *hunt of the bear or *the bears' hunt.