. I was told after 90 days my pay rate would go from $10.10 to $10.60
Yeah, so, whoever told you that was either horribly out of date with their information, or flat-out lied.
And, it won't.
How are you at time traveling? You could go back in time at least 2.5 years to when the previous union contract was in effect, it provided for an increase in pay rate after 90 days. Of course, you would have started at $8.50 and gone up to $9.00 under that contract.
Actually, the current contract only calls for $10; you get an extra 10¢ because the POTUS raised the minimum wage for federal contractors. Thanks Obama!
I suggest you get a copy of the UPS-Teamsters contract or look it up online at the Teamsters website. Part time wages are in Article 22 (Section 5, I believe).