When get a new computer, your iTunes of cause is needed to reinstalled on it. Consequently, your iTunes library is empty and will wipe out your iPhone library then leave you no library anywhere. So if you want to tansfer files from from iPhone to New Computer, you'd better turn to some way else. You can try some third-party transfer software, like Cucusoft iPhone to Computer Transfer which is powerful in iPhone backup software.

The full version of iPad/iPhone to Computer transfer can:

Recover, backup, restore iPhone files
Transfer files between iPods, iPads and iPhones
Transfer iPhone songs, videos, photo, podcast playlis from your iPhone (any model) back to your pc/itunes.

Guide: How to transfer files from iPhone to a new computer ?

Step 1: Connect iPhone to new computer and Launch iPhone to Computer transfer. Then the program will search the device automatically. If the device is enabled, the files (songs/movies/photos) will be shown in the list.

Step 2: Select files to transfer from iPhone to new computer, then Click the Export button.

Step 3: Choose transferring music from iPhone to new computer. After you clicking the Export button, the following window will pop up, you can choose transferring to computer or iTunes directly. You can set the output folder at Advanced->Options before transferring...

Step 4: When the transferring finished, the following window will pop up. It shows that how many files have been done. You can click the OK button to view the files transfered from iPhone to new computer.

Now, you have transfered files from iPhone to new computer, just enjoy it.

Some useful guides you can see here:
how to transfer music from ipod to ipod
how to transfer music from ipod to computer