I think it should always be "contact me" as contact is a transitive verb and should be followed by an object. But could someone clarifies? Is "contact with me" correct usage in some context?
When used as a verb, "contact" is always transitive. As such, you are correct: it should always be "contact me."
Note that the phrase "contact with me" may appear as an object in other situations, where it may be mistaken for a verb. Example: The other day, my long lost brother made contact with me. In this example, made is the verb, and contact is its direct object.
"to contact" is mostly handled as a transitive verb. "to contact with me" seems to be occasional use as Ngrams shows:
"to contact with sb" was in use long before "to contact sb". Maybe it was influenced by "get in contact with me". Later the simple transitive version got the better of it.