Two-proton decay of the $^{17}\mathrm{Ne}$ low-lying states populated in the $^{1}\mathrm{H}(^{18}\mathrm{Ne},d)^{17}\mathrm{Ne}$ transfer reaction is studied. The two-proton width ${\mathrm{\ensuremath{\Gamma}}}_{2p}$ of the $^{17}\mathrm{Ne}$ first excited $3/{2}^{\ensuremath{-}}$ state at ${E}^{*}=1.288$ MeV is of importance for the two-proton radioactivity theory and nuclear-astrophysics applications. A dedicated search for the two-proton emission of this state was performed leading to the new upper limit obtained for the width ratio ${\mathrm{\ensuremath{\Gamma}}}_{2p}/{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\gamma}}l1.6(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$. An original, ``combined mass'' method is suggested and tested as capable of improving the resolution of the experiment, which is a prime significance for the study of nuclear states with extremely small particle-to-gamma width ratios ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{part}}/{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\gamma}}$. The condition ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{part}}\ensuremath{\ll}{\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\gamma}}$ is quite common for the states of astrophysical interest, which makes the proposed approach promising in this field.