Given an irreducible sofic shift
X
X
, we show that an irreducible shift of finite type
Y
Y
of lower entropy is a factor of
X
X
if and only if it is a factor of
X
X
by an open bi-continuing code. If these equivalent conditions hold and
Y
Y
is mixing, then any code from a proper subshift of
X
X
to
Y
Y
can be extended to an open bi-continuing code on
X
X
. These results are still valid when
X
X
is assumed to be only an almost specified shift, i.e., a subshift satisfying an irreducible version of the specification property.