A certainty after you refinance or get a new home loan is that your mailbox will be full of offers for bad products.
First you will see offers for mortgage insurance. This is expensive term insurance and you are much better off buying a level term policy in the open market. Also, most people will need much more than the value of their mortgage as a death benefit.
Second is the biweekly mortgage programs. The idea is that if you make a mortgage payment equal to half your monthly payment every two weeks, instead of once a month, you end up with one extra payment per year. The pitch is that you will have your mortgage paid off in about 22 years instead of 30 years. What's the problem you might ask? Well, the soliciting company will charge a fee of a couple hundred dollars to facilitate this additional payment when anyone with a checkbook or online access to their lender could just go in and make one extra payment per year at no cost. Even if you can afford the extra payment per year, it's not necessarily a great idea.
If you are comfortable with investments and the inherent risk, you could invest that extra payment and if your return is higher than your after tax cost of the mortgage, you would end up with more money at the end of the loan. If I have a mortgage at 3.25%, after the federal tax deduction*, that mortgage costs right around 2%. If my investment return exceeds 2%, I come out ahead. If rates increase on new mortgages, the advice could change but in today's current low rate environment, the investment return needed is a fairly low hurdle. I recently received a solicitation with a 7% illustrated rate but I'm not sure of anyone that has a rate that high in today's environment.
* estimating 40% tax rate