Another mortgage question!

My 5 year fixed rate has just come to an end and I have been moved onto a variable rate of 2.5% (2% above base rate).

The mortgage runs for approx 16 more years but I want to increase the payments to pay it off faster. I am going to start making bigger payments to achieve this but my question is:

I realise that making extra payments will shorten the life of my mortgage but should I be asking the bank to actually reduce the term of the mortgage - would that make a difference to the overall total amount paid? (I believe there would also be a fee to shorten the term.)