Curious to know why you can't take both for a walk. I used to carry Wilf around in a pouch socializing him while I took Ruby for a walk.
Me too with Misty, but without the pouch
. I tried to alternate between taking her out and leaving her at home, to get her used to being alone. In the park I found other dog owners were fighting to have a go at carrying Misty, which was good for socialisation too.
This meant I could still give Jimmy loads of attention and he didn't get jealous of the new pup.
If you are concerned about the new pup catching something from Millie, couldn't you wash Millie's paws when you get home ?