The logarithmic correction for the order of the maximum for two-speed branching Brownian motion changes discontinuously when approaching slopes $σ_1^2=σ_2^2=1$ which corresponds to standard branching Brownian motion. In this article we study this transition more closely by choosing $σ_1^2=1\pm t^{-α}$ and $σ_2^2=1\pm t^{-α}$. We show that the logarithmic correction for the order of the maximum now smoothly interpolates between the correction in the iid case $\frac{1}{2\sqrt 2}\ln(t),\;\frac{3}{2\sqrt 2}\ln(t)$ and $\frac{6}{2\sqrt 2}\ln(t)$ when $0<α<\frac{1}{2}$. This is due to the localisation of extremal particles at the time of speed change which depends on $α$ and differs from the one in standard branching Brownian motion. We also establish in all cases the asymptotic law of the maximum and characterise the extremal process, which turns out to coincide essentially with that of standard branching Brownian motion.