Two-proton decay of the Ne-17 low-lying states populated in the H-1(Ne-18, d) Ne-17 transfer reaction is studied. The two-proton width Gamma(2p) of the Ne-17 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) x 10^(-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 Gamma_(part)/Gamma_(gamma). The condition Gamma_(part) << Gamma_(gamma) quite common for the states of astrophysical interest, which makes the proposed approach promising in this field.