  Stochastic resetting has emerged as a useful strategy to reduce the
completion time for a broad class of first passage processes. In the canonical
setup, one intermittently resets a given system to its initial configuration
only to start afresh and continue evolving in time until the target goal is
met. This is, however, an instantaneous process and thus less feasible for any
practical purposes. A crucial generalization in this regard is to consider a
finite-time return process which has significant ramifications to the first
passage properties. Intriguingly, it has recently been shown that for diffusive
search processes, returning in finite but stochastic time can gain significant
speed-up over the instantaneous resetting process. Unlike diffusion which has a
diverging mean completion time, in this paper, we ask whether this phenomena
can also be observed for a first passage process with finite mean completion
time. To this end, we explore the set-up of a classical drift-diffusive search
process in one dimension with stochastic resetting and further assume that the
return phase is modulated by a potential $U(x)=\lambda |x|$ with $\lambda>0$.
For this process, we compute the mean first passage time exactly and underpin
its characteristics with respect to the resetting rate and potential strength.
We find a unified phase space that allows us to explore and identify the system
parameter regions where stochastic return supersedes over both the underlying
process and the process under instantaneous resetting. Furthermore and quite
interestingly, we find that for a range of parameters the mean completion time
under stochastic return protocol can be reduced further than the
\textit{optimally restarted} instantaneous processes. We thus believe that
resetting with stochastic returns can serve as a better optimization strategy
owing to its dominance over classical first passage under resetting.
