Ambient backscatter communication (AmBC) has been introduced to address communication and power efficiency issues for short-range and low-power Internet-of-Things (IoT) applications. On the other hand, reconfigurable intelligent surface (RIS) has been recently proposed as a promising approach that can control the propagation environment especially in indoor communication environments. In this paper, we propose a new AmBC model over ambient orthogonal-frequency-division-multiplexing (OFDM) subcarriers in the frequency domain in conjunction with RIS for short-range communication scenarios. A tag transmits one bit per each OFDM subcarrier broadcasted from a WiFi access point. Then, RIS augments the signal quality at a reader by compensating the phase distortion effect of multipath channel on the incident signal. We also exploit the special spectrum structure of OFDM to transmit more data over its squeezed orthogonal subcarriers in the frequency domain. Consequently, the proposed method improves the bit-error-rate (BER) performance and provides a higher data rate compared to existing AmBC methods. Analytical and numerical evaluations show the superior performance of the proposed approach in terms of BER and data rate.