Can we contract the sentence
Amy has had many different jobs
to this one?
Amy's had many different jobs
Yes, you certainly can, and in speech most people would.
Omitting "has" changes the meaning slightly, because it changes nthe tense. It is not a simple contraction or elision.
Amy has had many different jobs.
This is in the Present Perfect tense. In this construction it indicates that Amy has had many jobs over a period starting in nthe past, and continuing into the present, or over the course of her entire career to date.
Amy's had many different jobs.
This is too easily confused with
Amy had many different jobs.
This third example is in the simple past, and implies thatm there was a period when Amy had many jobs, but this period is now over, and ended soem time in the past.
The second example, using the "'s" could be considered a simple contraction of the first. But I would advise against using it, because it obscures the "has" ,that is the marker of the present perfect, and is too easily confused.
It is true that even between the first and third example the difference in meaning is slight and subtle, and in many cases would not matter.