Be specific about the loan you want. The more precise you are, the easier it will be for a loan officer or mortgage broker to find the best rate. For example, do you want a 15-year or 30-year mortgage? Do you need a "jumbo" mortgage (more than $333,700)? Are you willing to pay points to reduce the interest rate? The answers to these questions will depend on several factors, including when you plan to sell the house and how soon you want to retire your debt. Only you know the answers, but the sooner you know them, the better.
If you are able to get a lower rate that what you currently have, you can save tens of thousands of dollars over the life of your loan. Also, most lenders don't charge as many fees to refinance a mortgage and depending on how much equity you have in your home you may be able to roll the closing costs into your new loan, still have a lower balance than your original loan, a lower rate, and a lower payment.
Ask the company carrying your current title insurance policy what it would cost to reissue the policy for a new loan. This may reduce your cost.
If your property value reduces and you refinance up to 80% of the reappraised value, your original mortgage amount may be higher than this amount. Thus, the new loan will not be sufficient enough to help you pay down the existing one.