If you fix a prime $\ell$, and consider the Galois action of the decomposition group $D_p$ on the $\ell$-adic Tate module, then (in a standard way, due to Deligne) you can convert this action into a representation of the Weil--Deligne group, and so in particular of the Weil group. Restricting to the inertia group, you get a representation of the inertia group $I_p$, known as the inertial type $\tau$. It is independent of $\ell$. (The only reason to detour through the Weil--Deligne group is to deal with possibly infinite image of tame inertia; if the elliptic curve has potentially good reduction, then we can skip this step and just take the representation of $I_p$ on the $\ell$-adic Tate module, which has finite image and is independent of $\ell$.)
Now one can look at the deformation ring $R_{\rho}^{[0,1],\tau}$ parameterizing lifts of $\rho$ of which at $p$ are of inertial type $\tau$ and Hodge--Tate weights $0$ and $1$. (See Kisin's recent JAMS paper about potentially semi-stable deformation rings.)
Let's suppose that $E$ really does have potentially good reduction. Then Kisin's "moduli of finite flat group schemes" paper shows that any lift parameterized by $R^{[0,1],\tau}$ is modular. This shows that $R^{[0,1],\tau} = {\mathbb T},$ for an appropriately chosen ${\mathbb T}$.
One thing to note: unlike in the original Taylor--Wiles setting, from this statement one doesn't get quantitive information about adjoint Selmer groups, and one doesn't get any simple interpretation of what this $R = {\mathbb T}$ theorem means on the integral level. (In other words, Artinian-valued points of $R^{[0,1],\tau}$ have no simple interpretation in terms of a ramification condition at $p$.)