  Two-proton decay of the $^{17}$Ne low-lying states populated in the
$^1$H($^{18}$Ne,$d$)$^{17}$Ne transfer reaction was studied. The two-proton
width $\Gamma_{2p}$ of the $^{17}$Ne first excited $3/2^-$ 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 $\Gamma_{2p}/\Gamma_{\gamma} < 1.6(3) \times 10^{-4}$. A
novel, "combined mass" method is suggested and tested capable to improve the
resolution of the experiment which is a prime significance for the study of
nuclear states with extremely small particle-to-gamma width ratios
$\Gamma_{\mathrm{part}}/\Gamma_{\gamma}$. The condition $\Gamma_{\mathrm{part}}
\ll \Gamma_{\gamma}$ is quite common for the states of astrophysical interest
which makes the proposed approach promising in this field.
