In this case, it's better to cancel your subscription before updating your payment info. You will still be Premium until the remaining days of your subscription are used up.
What I'm suggesting you to do is this:
1. Go to your Subscription page.
2. Click UPDATE under Payment Info. If it shows below 'Would you like to cancel your subscription?' click there and cancel.
3. Check the box to 'Remove', and click 'Change' button,
Note: If you have your Premium active you won't be able to remove your payment info. That's why it's important to cancel before removing your payment info.
After you've removed your payment info, resubscribe to Premium with the new card info to ensure that your offline music won't be deleted and that your subscription will renew correctly when it's time.
Hope it helps. 🙂
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Note: I'm not a Spotify employee.