The boundary modified contact process models an epidemic spreading in one dimension with two infection parameters, $λ_i$ and $λ_e$. Starting from a finite infected set, each edge of $\mathbb{Z}$ transmits the infection at rate $λ_i$ except for the rightmost and leftmost edges incident to infected vertices, which transmit the infection at rate $λ_e$. We show a strong law of large numbers and central limit theorem for the location of the rightmost infected vertex when $λ_i = λ_c$ and $λ_e = λ_c + \varepsilon$. We also show stretched exponential tail bounds in the fluctuations of the rightmost infected vertex, the extinction time of the process on the event of non-survival, and the probability of survival given the size of the initial infected region. Our results extend to the boundary modified contact process whenever $λ_c \leq λ_i < λ_e$, and solves an open problem first proposed by Andjel and Rolla in [1].