How to download previously purchased iTunes songs

The iTunes store has a 'Purchased' section that you can use to view and re-download purchased music.

Open up iTunes on your computer and select iTunes Store in the sidebar. In the iTunes Store homepage, click the Purchased link in the Quick Links box on the right:

Once on the Purchased page, select the Music tab on the left and the 'Not on This Computer' tab on the right. Select All Songs in the list on the left and click the Download All button at the bottom of the page.

Once your downloads finish you should have all your music in your library, ready to be synced to your new iPhone.

I had this exact same problem and I tried the same solution you did: deleted the file and could not redownload. It just said "Purchased" and there was no option to redownload. It turns out that I must have selected to "hide" the purchase. I fixed this by going to my account settings in itunes and clicking "view hidden purchases". Low and behold, my missing file was there. I clicked the "unhide" button and was able to download it again from the itunes store.