I model the survival data with a piece wise constant exponential distribution at time t. Let R be the total number of the population at time t, and D be the number of deaths observed at time t+dt. So, we may estimate the local hazard function as D/dt from the Poisson process or as D/R from the Kaplan-Meier model. Could we make connections between the two formulas or am I wrong somewhere?
You probably mean that you assume a piece-wise exponential distribution of survival times, which implies a piece-wise constant hazard function. The respective model is called the Piece-wise exponential model (PEM).
There is a connection between the PEM and Cox PH, in fact they are equivalent in some situations (see example here and the Whitehead 1980 reference therein).
However, it is usually better to use an extension of the PEM, the Piece-wise exponential additive (mixed) model (PAMM) that can be estimated using GAMMs.
I wrote an R package
pammtools specifically to do so.
See this vignette to get started with PEM and PAMM: https://adibender.github.io/pammtools/articles/baseline.html
In the articles section there are many more examples (and comparisons to other models, e.g.