Recently, data hiding techniques have become very popular in several vital applications, especially in telemedicine. The reason for this is their ability to give good results such as high embedding capacity while preserving visual image quality as much as possible after extracting the hidden secret message. In earlier studies, many researchers have achieved the goal of reversible data hiding (RDH) algorithm. All these methods have achieved excellent results on standard and natural images. However, in the case of medical images, especially color medical images, we face the problem of how to preserve the visual quality of image contents while achieving the goals of RDH in avoiding the loss of patient data or the distortion of the diagnosing image. In this paper, we proposed a secure data hiding method using a hyper chaotic map and left-most embedding strategy. The proposed methods are hybrid, where it is applied in the DCT frequency domain and encrypted domain together as presented here. This gives a higher embedding rate and higher visual image quality than existing methods without any loss or distortion of both hidden secrets data and reconstructed image. The novelty of this paper is to embed the desired secret data in each quantized block of DCT using (8-bit LMSB) strategy for embedding process. We tested our algorithm on both color medical images and standard color images of different sizes and different formats. We evaluated the performance of our algorithm on the basis of the quality metrics MSE, PSNR, BER, SSIM, Correlation, Symbol Error Rate, additional quality evaluation metrics, execution time, and different types of geometric and signal attacks. All of these parameters are demonstrated and represented in this proposed work in detail.