Here's a suggestion that you may or may not have done. Don't present it like church visitation just happens 7:00 on Tuesday nights. Then when the person can't come, everybody feels bad.
My church does have a FAITH team on Tuesdays. And you are right, that only a small percentage of people go to that. But just like many other Baptist churches, there is some important meeting almost every night.
And there are many Saturday projects.
We have a lot of visitation though. Through cards to take and email, we can get visiting assignments to do on the way home from work on Thursday afternoon, Monday morning, whenever it is going to work for you and your target visit.
There are elderly people in our church who don't drive at night but do visitation for the church on the phone.
More visitation goes on than you realize, I imagine. Over the years I have witnessed to my neighbors extensively, yet it never showed up on my church's visitation statistics. I have done visitation at Walmart and taking people to lunch. That didn't show up on the stats either.
And official visitation is a method that does not work in every region of the country. Gated communities and apartment complexes often prevent it. Having said all that, I think visitation is important. But sometimes if there isn't enough, some other church meetings need to be cancelled to make time. (Only a general comment.) I don't imply anything specific about your churches.
Karen