Individual lenders have a lot of leeway to offer you a better rate in order to secure your business. When I was shopping for a mortgage last year, every single person I talked to said they would beat any other written offer I was given (spoiler alert: they lied).
So I took them up on that. I called like six places (from bankrate and lendingtree) and spent about half an hour with each one explaining that I was rate shopping and then getting the details of their offer for the home I wanted to buy.
Then I called back the place that had given me the best deal and told them they were the best I'd seen so far, so they should send me a written offer. They do this all the time, it's just a form letter with the numbers on it. I took that offer to the other lenders and said here's a better written offer than you gave me, can you beat this or not?
Some of them said yes and gave me a better rate than they had the day before. Most of them tried to BS me with nonsense about how they were going to provide me superior service that warranted their higher rate, but really they only work on my loan for like a day and then they collect my payments for 15 years, so I wasn't about to pay extra for that crap. You have to really cut through the sales pitches about APR vs APY vs closing costs vs fees, and even then you're never going to be 100% certain what it will cost you because they always find ways to hide costs from you. A mortgage is a complicated financial instrument and there are lots of different ways to skim from it.
I took the new best offer and repeated the process with the people who were still answering my calls. Some didn't, some dropped out, a few told me flat out that they couldn't possibly beat the offer I already had in-hand. At about that point I figured I was getting close, so I called back the one other guy who was still trying really hard to convince me that he wanted my business at all costs. He was adamant that he would absolutely beat any other offer. He looked at the offer and told me he couldn't beat it.
So I locked in my rate the next day, with a private local mortgage lender/broker. It was a busy three or four day of phone calls and emailing, but I lowered my rate at least half a point and so saved myself many tens of thousands of dollars over the life of my loan. Like anything else in life, a little education and hard work can save you money. I figure I wouldn't buy a refrigerator or washing machine without doing at least a full day of research, and those cost only a tiny fraction of what my mortgage does so it's probably worth the effort.