We present results of several measurements of CsI[Na] scintillation response to 3-60 keV energy nuclear recoils performed by the COHERENT collaboration using tagged neutron elastic scattering experiments and an endpoint technique. Earlier results, used to estimate the coherent elastic neutrino-nucleus scattering (CEvNS) event rate for the first observation of this process achieved by COHERENT at the Spallation Neutron Source (SNS), have been reassessed. We discuss corrections for the identified systematic effects and update the respective uncertainty values. The impact of updated results on future precision tests of CEvNS is estimated. We scrutinize potential systematic effects that could affect each measurement. In particular we confirm the response of the H11934-200 Hamamatsu photomultiplier tube (PMT) used for the measurements presented in this study to be linear in the relevant signal scale region.