Using gift cards back to back?

[andreastunes]

For Christmas I got 2 Spotify Premium gift cards for $60 each. I ended up activating the first one and I got an email this morning that it was going to expire on June 30th. I went online to activate my second one (for the next 6 months) and it says that it will begin whenever the payment is processed.

So my question is, will my first gift card "run out" before this one starts to be used? (Aka, will I still be using up my first gift card until the 30th) Or should I have waited until that one fully expired before activating this one? Because if this one starts being used now I will have lost a week of my subscription time. (It will expire December 23rd instead of December 30th).