In most cases, with most airline situations, you're not going to have a speedy upgrade. You're going to get time in the right seat regardless of your experience or background, as you'll be restricted from upgrading by seniority. By the time you do get to upgrade, you'll have your thousand hours.
The reality is that you should have that time and experience first, in order to act as pilot in command, so take advantage of getting it. See how the operation works, how it's conducted, so when you do eventually slide into the left seat, it's a familiar fit with that company and aircraft.