In April, 2011 I quit my job in Chicago and moved all my belongings to Phoenix and moved in with a family member while I looked for work. I could only find temporary work for a short term situation (4 months) so I moved back to Chicago in November, 2011. I started working as a contractor for my current employer in December, 2011 becoming a permanent employee in May, 2012 and still work there.

The expenses for the combined moves by two different moving companies exceed $8,000. Can I deduct the expenses for only one of the moves or both?

You would only be able to deduct moving expenses for the second move if
You meet the time test for that one.
Here are the requirements for the time test from publication 521:
Publication 521 (2012), Moving Expenses

Time Test for Employees

If you are an employee, you must work full time for at least 39 weeks during the first 12 months after you arrive in the general area of your new job location (39-week test). Full-time employment depends on what is usual for your type of work in your area.

For purposes of this test, the following four rules apply.

You count only your full-time work as an employee, not any work you do as a self-employed person.

You do not have to work for the same employer for all 39 weeks.

You do not have to work 39 weeks in a row.

You must work full time within the same general commuting area for all 39 weeks.

For sure, the first move is NOT deductible, and the second move would be a hard sell to the IRS in an audit because you moved BACK to Chicago.

A better approach would be to claim the Phoenix move as a temporary job move and claim the expenses for the trip on Form 2106 as an itemized deduction.