A tiled web map provides just a series of tile bitmaps at fixed zoom levels.
I am not aware of any tile server that allows to client to customize its rendering styles.
You could try some different tile server that is configured to render more details at a specific scale, or implement your own tile server.
But as long as you're using the same server, the only way to increase the level of detail is to increase the resolution, i.e., to use a bigger monitor (and then the text labels will appear smaller).
(This is how all tile servers work; it is not specific to either OSM or MapInfo.)