  We prove automorphy lifting results for certain essentially conjugate
self-dual $p$-adic Galois representations $\rho$ over CM imaginary fields $F$,
which satisfy in particular that $p$ splits in $F$, and that the restriction of
$\rho$ on any decomposition group above $p$ is reducible with all the
Jordan-H\"older factors of dimension at most $2$. We also show some results on
Breuil's locally analytic socle conjecture in certain non-trianguline case. The
main results are obtained by establishing an $R=\mathbb{T}$-type result over
the $\mathrm{GL}_2(\mathbb{Q}_p)$-ordinary families considered by Breuil-Ding.
