good cover tree?

I haven't been on here for AGES but am back with a question if you don't mind please.

Towards the bottom of our garden we have a large bay tree 'mass'. Not sure quite how to describe it, it must be numerous trees and bushes and has been there a very long time. It smells now and we want to get it all chopped down and taken out. HOWEVER, we like trees and we do want to keep some privacy so we want to plant some new trees a bit further down the garden (there is a big unused gap of about 6ft or more between the bay mess and the wall at the bottom of the garden) What we are thinking is getting a couple of quite mature evergreen trees to provide the cover once it is taken out.

What sort of trees are evergreen, provide good privacy cover, look attractive, not too high maintenance and not too expensive to get at a reasonable size?