There's some tension in the desire to beat your own path away from from the main tourist itinerary, and the desire for good rail connections. As someone else mentioned upthread, to get the flavor of rural Shikoku you really need to get out of the big cities. I would go further to say that you need to travel to places that aren't well served by trains.

How to overcome this contradiction? ... Bring a folding bicycle, maybe. I am thinking of investing in a Brompton for pretty much that purpose. Those bicycles may give you sticker shock , but a tiny bike that rides well and that's solid enough to handle a bit of touring might be a transformative piece of kit.

There's also JR rental cars at some stations...