Either turn the phone off or eject it, then put the new one in.
If you want to keep all the data on the new one that was on the old one, you'll have to either copy from one to the other if you have th hardware to have them both plugged into a computer at the same time, or plug in the old one, copy all the data to the PC, safely eject it, plug in the new one, copy all the data from the PC to the new one, safely eject the new one and put it into the phone.
Note that if you have apps "moved" to the old card, you'll have to move them back to internal storage first, then "move" them back to the new card. You can't copy them from one card to the other. (And if you're running Android 6, the cards have to be formatted as "external". If you formatted the current one as internal, you can't copy files from it using a PC.)