What is the preferred order of operations when replacing a mezuzah?

Question

The scenario: While on vacation I bought a beautiful new mezuzah case, it is small but quite nice. The plan is the use it to replace a mezuzah already in the house. There are lots of sources saying this is an acceptable thing to do. This is the first one that came to hand/Google. The new mezuzah is sufficiently smaller than the old one to need a new scroll. So there are two ways to go about doing the replacement:

1) Remove the old mezuzah completely then put up the new one.

2) Put up the new one in the available space then take down the old one.

Are there any sources that distinguish between the two order of operations and give a preference?

I am working on the assumption that both are halachically acceptable but I am curious if there is any basis for a preference for one or the other.

Answer 1

The Pitchei Teshuva (YD 291 sk 2) quotes the Chamudei Daniyel that attaching a second Mezuzah to the same doorpost violates Bal Tosif and would thus be a biblical prohibition. Accordingly you should strongly prefer (unless your life is threatened) taking the old Mezuzah down first.