We have proposed and demonstrated a weak acoustic signal detection technology based on phase-sensitive optical time-domain reflectometry (Φ-OTDR). Non-contact acoustic signals transmitting through air gap between the sound source and the receiver are difficult to detect due to fast attenuation. In order to improve the detection ability of non-contact weak acoustic signals, we demonstrate that multi-mode fiber (MMF) is a better solution than single-mode fiber (SMF) benefiting from its larger core and higher Rayleigh backscattering (RBS) capture coefficient. The frequency signal-to-noise ratio (SNR) has been enhanced by 9.26 dB. Then, with the help of 3D printing technology, elastomers have been designed to further enhance the detection ability due to the high-sensitive response to acoustic signals. Compared with the previous reported "I" type elastomer, the location and frequency SNR enhancement caused by our new proposed "n" type elastomer are 8.39 dB and 11.02 dB in SMF based system. The values are further improved to 10.51 dB and 13.38 dB in MMF and "n" type elastomer integrated system. And a phase-pressure sensitivity of -94.62 dB re rad/µPa has been achieved at 2.5 kHz. This non-contact weak acoustic signal detection technique has great application potential in the quasi-distributed partial discharge (PD) detection of smart grid.