Although the legal limit on tread is 1.6mm, 3mm is about as low as I tend to go. So getting at least three new tyres is probably a good move. When I had Michelin tyres, they were fine at first, but they got very noisy when getting towards the limit of wear.
Alternatively, you could change all four for the newer all-season tyres, not the old ones with just a coarse tread, but the ones that are winter rated, like the Michelin Cross Climate. I fitted a set of Goodyear Vector 4season tyres. They have performed well over the last 2½ years, and are lasting well enough, even though I don't do somany miles now. They have travelled abroad and survived higher speeds. Look for the "mountain & snowflake" symbol on the sidewall.