If you think you are going to rent, and need to refi, you should do it now while you are in the house, so you can get the low rate.
Time frame from application to signing -- 6-8 weeks.
Refi also costs money, a few thousand in fees so add that in as an expense. If you have not paid off 20% of your current mortgage, you might have PMI added onto your new mortgage cost.
You can add all these extra fees into your new mortgage, but it will increase your new mortgage also.
Other costs to consider in your rental plan --
who pays for the utilities (gas/oil or water (if you get a town bill))
Replacement of light bulbs (interior or exterior)!
Trash responsibility/costs (our town, we have to have special bins and bags per week that we pay for).
Landscaping -- who is responsible, who pays or you hire someone for a price and factor that into the rent too.
Snow removal responsibility -- if you have to add that in.
House insurance on a rental -- sky high so you will have to go out and shop around first, get some quotes, and then factor that into your monthly fee. (insurance costs divided by 12 plus taxes divided by 12 PLUS add 10-20% for increases plus your mortgage payment (house only)= monthly payment)
Currently what you pay, is designed for you, but for a renter, the insurance and taxes will be higher to cover increased costs.
Another cost you may have to add in also, is if you have public/town water and you get bills for those, in our state, renters cannot be responsible to pay this, the home owner is, so this cost has to be factored into the monthly rent also.
Keep in mind that some people use LOTS of water (maybe more than you have used) so you have to factor in those costs also or else you will be paying out of your pocket the increased expense.
This would also matter if you are leaving behind a washer or dishwasher appliance (frequency of use could increase or decrease water used).
The lease with option to buy is a good idea.