He put on his coat and left the room.
Can I also say: He put his coat on and left the room. in other words, is there any rule for where to put the "on" or "off" or others from the phrasal verbs in the sentences??

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Transitive phrasal verbs are divided into separable or inseparable. Your dictionary should tell you which they are.

Separable phrasal verbs can put the object before or after the particle but if you use an *object pronoun * it MUST go before the particle.

He put on his coat
He put his coat on
He put it on

Never: "He put on it"


Inseparable phrasal verbs must always have the object after the particle, even if it's an object pronoun.

He came across a dog.
He came across it.

Never "He came a dog across."
Never "He came it across"