Open disk management, right-click the drive in question and the unused space and select "extend partition" or similar, depending on your OS it might be worded slightly different.
What happens when you clone (for example) a 120GB to a 250GB SSD, it clones the partition exactly as it was - so it's still a viewed as 120GB 'drive' but, in actuality, it's now a 120GB partition on a 250GB drive.
Alternatively, you can create a secondary partition with the remaining space...... but that kind of defeats the purpose of moving to a larger drive in the first place.