No, it doesn't have to be the same accommodation - as long as you can prove you've been renting privately for 6 of the last 12 months, that will be acceptable. It may be useful if you can show that you had planned the move before the news of the unemployment came through.
You don't state whether you're working or not - if you're working full-time, then your partner won't qualify for Rent Supplement regardless of your income.
If you're working part-time (under 30 hrs per week), then your income will be assessed (as well as any SW payment you or your partner has) to decide entitlement to Rent Supplement. There is also a maximum rent limit which applies.