What next?!

East Midlands front lawn, recently moved in and noticed a lot of moss in the grass. Gets good sunlight majority of the day as we are E/E/S facing.

Scarified, then hand raked, scarified on a lower setting and half way through hand raking but still a lot of moss in the grass.

Should I now go to the lowest scarify level and then seed or overseed over lawn now despite there still being moss.