This paper proposes a multiple code block transmission scheme using hierarchical modulation (HM) for a broadcast channel in the orthogonal frequency division multiplexing (OFDM) downlink. We investigate the average bit error rate (BER) performance of two-layer HM using 16 quadrature amplitude modulation (QAM) and three-layer HM using 64QAM in multipath Rayleigh fading channels. In multiple code block transmission using HM, the basic information bits are demodulated and decoded to all users within a cell that satisfy the bit error rate (BER) requirement. Hence, we investigate non-uniform QAM constellations to find one that suppresses the loss in the average BER of the basic information bits for HM to a low level compared to that using the original constellation in which only the basic information bits are transmitted while simultaneously minimizing the loss in the average BER of the secondary and tertiary information bits from the original constellations in which the information bits of the respective layers are transmitted alone. Based on the path loss equations in the Urban Macro and Rural Macro scenarios, we also investigate the maximum distance from a base station (BS) for the information bits of each layer to attain the required average received signal-to-noise power ratio (SNR) that achieves the average BER of 10-3.