Hi Rolutry ,
Have you used the iTunes or iCloud to backup your iPhone 5s contacts or any other data?Maybe you can transfer the data to your new iPhone SE with the iTunes.
Or you can have a try with the software iSkysoft Phone Transfer which is my friend recommended to me. I have tested in my iPhone 6, just one click can solve your problem. The picture below is the interface of the software, it is clear to see that you just need to select the contactd , the call logs or other contents then click the Start Copy button.So easy and so useful!
[image|809042]