This is a general explanation, eerste is a special case because it ends in an unstressed -e, see my post below.
You are right jpippoj. It would also be e.g. haar mooie kind and het mooie kind in contrast to the undetermined een mooi kind.
"An -e is also added if there is a demonstrative or possesive pronoun instead of a definite article". So in this case it is a possesive pronoun. I copied the quote from the full explanation on adjectives here.
But on second thought…
- een mooie man
- een mooi huis
- een eerste man
- een eerste huis
The last two aren't very common constructions, but it looks like they indicate that ordinals always get the extra -e, can anyone enlighten?
I'm not a native speaker, but I assume that "eerste" always ends in "e" simply because its uninflected form is already "eerste", and it therefore doesn't get an extra "-e" for "man" in this case (i.e. the inflected and uninflected form are the same). In this case it also helps to distinguish "eerste" from the word "eerst" ("first" as in "at first"), which is apparently not an adjective.
I also found the following literature on the topic:
You're right, I was a bit quick in answering and didn't pay attention to the specific adjective mentioned. Adjectives ending in an unstressed -e, don't get any ending. Examples are all ordinal numbers: zesde, tiende, drieënzestigste, honderdste, driehonderduizendvierhonderdtwaalfde.
Also see the explanation here