If you're truly ready to pay an extra $1000 every month, and are confident you'll likely always be able to, you should refinance to a 15 year mortgage.
15 year mortgages are typically sold at around a half a point lower interest rates, meaning that instead of your 4.375% APR, you'll get something like 3.875% APR. That's a lot of money over the course of the mortgage. You'll end up paying around a thousand a month more - so, exactly what you're thinking of doing - and not only save money from that earlier payment, but also have a lower interest rate. That 0.5% means something like $25k less over the life of the mortgage. It's also the difference in about $130 or so a month in your required payment.
Now of course you'll be locked into making that larger payment - so the difference between what you're suggesting and this is that you're paying an extra $25k in exchange for the ability to pay it off more slowly (in which case you'd also pay more interest, obviously, but in the best case scenario). In the 15 year scenario you must make those ~$4000 payments. In the 30 year scenario you can pay ~$2900 for a while if you lose your job or want to go on vacation or ... whatever.
Of course, the reverse is also true: you'll have to make the payments, so you will. Many people find enforced savings to be a good strategy (myself among them); I have a 15 year mortgage and am happy that I have to make the higher payment, because it means I can't spend that extra money frivolously.
So what I'd do if I were you is shop around for a 15 year refi. It'll cost a few grand, so don't take one unless you can save at least half a point, but if you can, do.