The regulatory requirements in cell processing, in the choice of a biomaterial scaffold and in quality control analysis, have to be followed in the clinical application of tissue-engineered grafts. Confirmation of sterility during quality control studies requires prolonged storage of the cell-based construct. After storage, preservation of the functional properties of the cells is an important prerequisite if the cells are to be used for cell-based tissue therapies. The study presented here shows the generation of 3D constructs based on Wharton’s jelly multipotent mesenchymal stromal cells (WJ-MSCs) and the clinically-acceptable HyaloFast® scaffold, and the effect of two- and six-day hypothermic storage of 3D cell-based constructs on the functional properties of populated cells. To study the viability, growth, gene expression, and paracrine secretion of WJ-MSCs within the scaffolds before and after storage, xeno-free culture conditions, metabolic, qPCR, and multiplex assays were applied. The WJ-MSCs adhered and proliferated within the 3D HyaloFast®. Our results show different viability of the cells after the 3D constructs have been stored under mild (25 °C) or strong (4 °C) hypothermia. At 4 °C, the significant decrease of metabolic activity of WJ-MSCs was detected after 2 days of storage, with almost complete cell loss after 6 days. In mild hypothermia (25 °C) the decrease in metabolic activity was less remarkable, confirming the suitability of these conditions for cell preservation in 3D environment. The significant changes were detected in gene expression and in the paracrine secretion profile after 2 and 6 days of storage at 25 °C. The results presented in this study are important for the rapid transfer of tissue engineering approaches into clinical applications.