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The study of adsorption of polymer chains from a solution onto a flat solid surface has been extensively investigated for more than 60 years, not only due to its relevance for potential technological and biological applications but also for its importance on many phenomena such as adhesion, surface coating, wetting, and adsorption chromatography, among others. A simple and useful coarse-grained polymer model for adsorption is the interacting self-avoiding walk (ISAW) with additional monomer-substrate interaction. We have performed extensive computer simulations by using an advanced contact-density chain-growth algorithm, in which the density of contacts can be directly obtained. As the number of configurations with a given number of surface contacts and monomer-monomer contacts is independent of the temperature and solvent conditions, the algorithm can be easily applied to get results for different solvent parameter values without the need of any extra simulations. ISAWs on cubic lattices with up to 503 monomers were simulated and, by considering corrections to scaling, the transition temperature and the crossover exponent were evaluated for several values of the solvent parameter. The study covers from the limit of super-self-avoiding walks, characterized by effective monomer-monomer repulsion, to poor solvent conditions that enable the formation of compact polymer structures. We have found evidences that the crossover exponent changes its value as the solvent conditions vary.