Consider the supercritical branching random walk on the real line in the boundary case and the associated Gibbs measure $ν_{n,β}$ on the $n^\text{th}$ generation, which is also the polymer measure on a disordered tree with inverse temperature $β$. The convergence of the partition function $W_{n,β}$, after rescaling, towards a nontrivial limit has been proved by Aïdékon and Shi in the critical case $β= 1$ and by Madaule when $β>1$. We study here the near-critical case, where $β_n \to 1$, and prove the convergence of $W_{n,β_n}$, after rescaling, towards a constant multiple of the limit of the derivative martingale. Moreover, trajectories of particles chosen according to the Gibbs measure $ν_{n,β}$ have been studied by Madaule in the critical case, with convergence towards the Brownian meander, and by Chen, Madaule and Mallein in the strong disorder regime, with convergence towards the normalized Brownian excursion. We prove here the convergence for trajectories of particles chosen according to the near-critical Gibbs measure and display continuous families of processes from the meander to the excursion or to the Brownian motion.