Matryoshka Words!

A Matryoshka doll is a set of wooden dolls of decreasing size placed one inside another. We're gonna do the same thing with words. Let's call it Matryoshka Words. A solution to this puzzle should contain every word in the chain. Each word in the chain adds a letter to the previous word's letters. The order of the letters does not matter.

For example, if I said top cards inside emptiness that would be: aces -> space.

Here is the definition of these Matryoshka Words:

A musical note inside a rodent inside a cover inside on the left side inside a door inside a meddler inside a bone inside officers inside a springboard.

Answer

The sequence seems to be:

ta — a note (a "ta" is a quarter note)
rat — a rodent

tarp — a cover
aport — on the left side
portal — a door
marplot — a meddler
temporal — a bone
patrolmen — officers
trampoline — a springboard