Why can’t I sign back into my App Store with my Apple ID after signing out?

What happens when you try? Error message(s)?

Do you have access to another WiFi network where you can try? That has worked for several people.

Try a restart.

Restart your iPhone, iPad, or iPod touch         

Try a forced restart.

Reset Device.      

If necessary

Hard reset.        

Do a backup and try a factory reset. Then restore from your backup.

Restore to Factory setting.      

Backup - Restore your device from an iCloud or iTunes backup.      

Maybe this read will help: If you can't connect to the App Store, iTunes Store, or other Apple services https://support.apple.com/en-us/HT201400