Q: Get song library from old phone to new phone

My old iPhone 5 has a full library of songs that includes many manually uploaded CDs. My old MacBook died. My new MacBook Air does not have those songs in iTunes. When I tried to backup my old iPhone so I could restore that music to my new phone, it does not actually back up the music (even though it says it backs up everything). I thought iTunes Match might work, but it says it does not match the songs on your device - only your computer. How do I get to those songs to my new phone? Air doesn't even have a CD player. Help!