  There has been considerable recent interest in type II string
compactifications on AdS$_3\times$S$^3\times$M$_{4}$ due to both new progress
in the AdS$_3$/CFT$_2$ correspondence and an independent search for new ${\cal
N}=(4,0)$ supergravity solutions. In this work we report a new family of small
${\cal N}=(4,0)$ preserving warped AdS$_3\times$S$^3\times $CY$_2$
compactifications of type IIB supergravity. The S$^3$ in this family is non
trivially fibered over the CY$_2$. We show how these new solutions provide a
IIB embedding of minimal ungauged $d=5$ supergravity coupled to an Abelian
vector multiplet. We also make a few initial steps towards the identification
of dual CFT$_2$ candidates. Specifically, we study the Kaluza-Klein spectrum of
the spin two and certain vector fluctuations that are expected to be dual to
the stress-energy tensor, SU(2) R current and several other BPS operator
families in the dual CFT$_2$.
