Transfer music iPhone to iPhone?

I had my iPhone 4 for 2 years and got a new iPhone 5s today. Everything else (apps, photos, contacts, notes, etc.) was restored in the backup - except my music. Now I don't always purchase songs from iTunes, and when I do, they're free. I like to download my songs off of other sites, then when I get them transferred onto iTunes and into my phone, I delete the file in my downloads folder to keep from taking up space. Does this mean I have to go back and redownload every single song? There are 300+ songs and I just need to know if this is going to be the hard or the easy way.

Hi, i want to say that we have the some situation. I have a ipad and i got i new iPhone 5C from my friends. I really don't want to download and buy the music again. I found AnyMP4 iphone transfer. I can't believe it can't only export ipad music to my new iphone, but also it can help me back up iphone photos, music, video, contact, sms, etc. On my computer for latter use. You can have a try. Really useful.

For the music, you can use the iTunes to sync to your new iPhone easily.