Actually Carcosan Page says "Any Phase", so you could use it during the OW Encounter Phase. However, I am under the impression that you cannot use card effects in the middle of an encounter unless they react specifically to such an encounter (like Tablet of Bast, Rerolls etc.)
Here's the relevant part from the FAQ:
Q: When can a character perform actions during a phase?
For example, if a player with the ally Duke (“discard to
immediately restore your Sanity to its maximum”) draws
an encounter and has to roll a die and lose that much
Sanity, when can he discard Duke to restore Sanity: not at
all once the encounter card is drawn, before the roll, or
after the roll?
A: Actions can be performed at any time during their
listed phase as long as any given conditions are met.
once an encounter is drawn, it must be resolved
before any more actions can be performed. So, in the
case of Duke, you can use him before or after drawing an
encounter card, but if you draw the encounter card and it
inflicts Sanity loss that reduces you to 0 Sanity, you go
insane before you can use Duke. Flesh Ward is one exception
to this rule because it prevents a Stamina loss and is
intended to be used in response to such events.