Yes, it's possible to start over with a brand-new account. However you need a second email address for this because the email address in the account you close can take up to 30 days before it becomes available. You can choose to start over with an empty account, or take some (or all) music with you.
(Skip the following paragraph if you want to start over with an empty account)
If there is any music that you'd like to take with you to your new account, you'll need to make sure it's saved in a Playlist. Music in Your Library that you want to take with you, will need to be placed in a Playlist. This is easily done on a computer as you can drag all songs at once from one Playlist to the other. When you've done this, you'll save the Playlists containing the songs you want to keep. You can do this by right-clicking a Playlist, going to Share and then clicking Copy Spotify URI. Save this link somewhere as we'll need it soon.
You can head over here and create a new account. Go through the process and log into the Spotify app. You'll have a brand new account with the same e-mail address!
Now is the time if you'd like to import your old songs. Paste the link you copied earlier in the search bar and open the Playlist you saved. From here you can place all these songs into one of your own Playlists or into Your Library on the new account.
The email address in your old account will become available within 30 days after you close the account. This means that you could change the email address in your new account to what it was in your old account. You can speed up with process if you reach out to support here.