The experimental study of the single, $A_\text{N}(t)$, and double, $A_\text{NN}(t)$, spin analyzing powers was carried out at The Polarized Atomic Hydrogen Jet Target polarimeter (HJET) at The Relativistic Heavy Ion Collider (RHIC). Large statistics of elastic $pp$ events accumulated during RHIC Runs 2015 ($E_\text{beam}=100\ \text{GeV}$) and 2017 ($E_\text{beam}=255\ \text{GeV}$) allowed us to measure analyzing powers in the CNI region $0.001\lesssim-t\lesssim0.02~\text{GeV}$ with unprecedented precision $|\delta A_\text{N,NN}|\lesssim2\times10^{-4}$. The hadronic single spin-flip $r_5$ and double spin-flip $r_2$ amplitude parameters were reliably isolated. Based on this two beam energies measurement, Pomeron and Regge-pole contributions to the hadronic single- and double- spin-flip amplitudes were evaluated. Extrapolation of the measured $r_5(s)$ to $\sqrt{s}=200\ \text{GeV}$ is in agreement with STAR measurements at this energy. A possibility to use HJET data to search for an Odderon contribution to the spin-flip amplitudes is discussed.