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In the present work, our main objective is to investigate the orbits of spinning test particles around a Schwarzschild black hole under the influence of a quintessence matter field (SQBH). We begin with the dynamics of the spinning test particles around SQBH which is governed by the Mathisson-Papapetrou-Dixon (MPD) equations under the pole-dipole approximation, where the gravitational field and the higher multipoles of the particle are neglected. Depending on the types of saddle points,the effective potential are classified and the possibility of chaotic orbits is discussed. The inner most stable circular orbits (ISCOs) of the spinning particle around SQBH are addressed, as are the effects of the parameters $S$ (particles' spin) and $ε$ (equation of state parameter). Later, Periastron precession is investigated up to the first-order spin correction for a spinning particle moving in nearly circular orbits around SQBH. It is noted that the addition of particle's spin revamps the results obtained for the non-spinning particles and also articulates the some interesting observational properties of the SQBH. Additionally, we discuss the ramifications of employing first-order spin corrections for analysing ISCOs, as well as compare our results to the Schwarzschild black hole to ensure that they are consistent in the limit when equation of state parameter $ε=-1/3$ and normalization factor $α\to 0$.