In visible light communications (VLCs) relying on intensity-modulation and direct detection (IM/DD), the conversion from electrical signals to optical signals and the limited dynamic range of the light-emitting diodes (LEDs) constitute the fundamental impediments in the way of high-integrity communications, especially when orthogonal frequency-division multiplexing (OFDM) is employed. In IM/DD VLCs, only real-valued positive signals are used for signal transmission. However, the Fourier transform of OFDM systems is operated in the complex domain. In order to meet the requirements of the IM/DD VLCs, the complex-to-real conversion is achieved at the cost of reducing the bandwidth efficiency. Moreover, OFDM signals experience a high peak-to-average power ratio; hence, typically clipping is used for confining the positive-valued signals within the LED's dynamic range. However, hard clipping leads to the loss of orthogonality for optical OFDM (O-OFDM) signals, generating inter-carrier interference. As a result, the performance of the clipping-based O-OFDM systems may be severely degraded. In this paper, the concept of piecewise companding transform (CT) is introduced into the O-OFDM system advocated, forming the CTO-OFDM arrangement. We first investigate the general principles and design criteria of the piecewise CTO-OFDM. Based on our studies, three types of piecewise companders, namely, the constant probability sub-distribution function, linear PsDF (LPsDF), and the non-LPsDF-based CT, are designed. Furthermore, we investigate the nonlinear effect of hard clipping and of our CT on O-OFDM systems in the context of different scenarios by both analytical and simulation techniques. Our investigations show that the CTO-OFDM constitutes a promising signaling scheme conceived for VLCs, which exhibits a high bandwidth efficiency, high flexibility, high reliability, as well as a high data-rate, despite experiencing nonlinear distortions.