A modified piece-wise polynomial pre-distortion is proposed, investigated and compared against classic memoryless polynomial pre-distortion. The proposed enhanced piece-wise polynomial pre-distortion uses the overlap between segments to ensure continuity in the total pre-distortion function. The implementation of overlap allows the use of a simple error estimation algorithm to minimise hand over error. The classic and proposed piece-wise pre-distortion performances are assessed and compared for several sets of polynomial coefficients. The proposed method is shown to consistently outperform classical polynomial pre-distortion in terms of required coefficients and linearity improvement. The method is applied for the linearisation of an envelope tracking Class-J PA at 1.7GHz, under a 1.4MHz LTE signal with a 14.4dB PAPR.