During the first phase of the deployment of new radio (NR) networks, base station (BS) and user equipment (UE) are expected to operate alongside the pre-existing long-term evolution (LTE) networks. This paper proposes a solution to be adopted by NR UE to perform an uplink transmission towards NR BS over the same time and frequency resources as the legacy LTE system, which would extend the uplink coverage when operating at the supplementary uplink band. In order to guarantee the absence of interference towards LTE BS, we design a new waveform termed as inverse discrete Fourier transform-based Vandermonde-subspace frequency division multiplexing (IDFT-VFDM) with the following important features. On one hand, an NR UE communicating with an NR BS through IDFT-VFDM would not generate any interference at the legacy LTE BS, regardless of the number of antennas present at both ends of the interference channel. On the other hand, an IDFT-VFDM signal can always be designed to occupy the same signal bandwidth as a legacy single-carrier frequency division multiple access signal for LTE uplink transmission. Numerical results demonstrate the merit of IDFT-VFDM and confirm its potential as a candidate solution to achieve LTE-NR co-existence in multi-antenna multi-user scenarios.