An unresolved issue in the study of baryon non-leptonic decays is that the theoretical values describing the s- and p-wave amplitudes of such decays cannot simultaneously accord well with experimental values. Compared with previous literature, this paper adopts the covariant chiral effective theory framework and calculates the one-loop corrections to the s- and p-wave amplitudes by using the extended minimal subtraction (EMS) scheme, and also takes into account the contributions from intermediate pion states that are neglected in previous studies (the contributions from intermediate decuplet states are not considered here). Unlike infrared regularization and the extended on-shell subtraction scheme, EMS is easier to implement and also avoids over-subtraction. Apart from the typical chiral logarithmic term m_slnm_s obtained in heavy-baryon formalism, the covariant calculation retains many non-local contributions that are not negligible. These non-local contributions vary with loop diagrams and intermediate states, making the complete covariant results significantly different from those from the simple chiral logarithmic structures in heavy-baryon formalism, which may alleviate the tension between the s- and p-wave components of the decay amplitudes. Subsequent numerical analysis confirms this conjecture. Two approaches are adopted to obtain covariant theoretical predictions: s-wave fitting and p-wave fitting. According to the fitted predictions and chi-squares of fitness, the s-wave fitting yields s-wave predictions slightly inferior to those under heavy-baryon formalism, but the resulting p-wave predictions are considerably improved compared with the heavy-baryon formalism predictions. The p-wave fitting produces p-wave predictions closer to experimental values, while the heavy-baryon predictions differ significantly from the experimental values. The resulting s-wave predictions from p-wave fitting show noticeable discrepancies with experimental data, but the heavy-baryon predictions are even worse. Therefore, working in the covariant framework, the tension between s- and p-wave amplitudes for baryon non-leptonic decays is significantly alleviated in comparison with that in heavy-baryon formalism. In addition, it is found that the contributions from intermediate pion states may be neglected in many cases, but are important and must be kept for decays with smaller experimental values.