Recently, asynchronous coarse-space correction has been achieved within both the overlapping Schwarz and the primal Schur frameworks. Both additive and multiplicative corrections have been discussed. In this paper, we address some implementation drawbacks of the proposed additive correction scheme. In the existing approach, each coarse solution is applied only once, leaving most of the iterations of the solver without coarse-space information while building the right-hand side of the coarse problem. Moreover, one-sided routines of the Message Passing Interface (MPI) standard were considered, which introduced the need for a sleep statement in the iterations loop of the coarse solver. This implies a tuning of the sleep period, which is a non-discrete quantity. In this paper, we improve the accuracy of the coarse right-hand side, which allowed for more frequent corrections. In addition, we highlight a two-sided implementation which better suits the asynchronous coarse-space correction scheme. Numerical experiments show a significant performance gain with such increased incorporation of the coarse space.