Made a 1 month contract into 4 month contract extension. How do I show my worth and make it full time?

Title says all. Got really lucky on getting my foot into the door working at HR at a major enterprise software company. How do I make this a full time opportunity?

I work as an assistant to the regional NA recruiting director during their major hiring season. I was also told that they are outsourcing my work to the Philippines. But I would like to take a chance on it.

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