It really depends on the temperaments of all the chicks, new and old.
This is a common issue during chick season and we get one or two threads, at least, per day on the subject of putting different aged chicks together. I've been there, too, and it's really a matter of trying it and seeing if it works. In my case, it didn't. The chicks were just too far apart in age and size.
However, if you ever think it would be dandy to add new chicks to a brooder of four-week olds, that would be the cut-off for simply saying no. By then the size difference is just too great.
You may be pushing it a bit to try three-week olds with new chicks, but you won't know until you try it. It was either last spring or the spring before that someone here wanted to put a single new chick with several three-week olds. We all said it was highly unlikely it would work. The OP updated their thread and said the new chick was a little tiger, actually bossing around the older chicks, more than holding its own with chicks more than double its size.
To increase your chances of it working, make sure you have a brooder of ample size so there's plenty of space for the new chicks to run away from the older ones. Also, the heat needs of the new chicks will be greater than those of the older ones. You may need to lower the heat lamp for the new chicks, if that's your heat source, and the older chicks will need plenty of space to get away from the heat.
Let us know how it works out.