I have a rather large button hole 3cm wide so it can accommodate a green part and I tuck it in. Still you need to find a carnation with a smaller green. I guess Astaire knew how to look for them. Back in the day when flower girls were selling carnations they must have had a different variety to what you can get now in a flower shop in terms of a size of a green part.
It is more difficult on a suit. I am struggling with a green part too. On an overcoat is fine. Also lightweight suit is not good for holding a carnation. It needs to be a proper weight.
Perhaps we have to go to our tailors with every jacket and make the button holes wider