Despite the extensive utilization of polysaccharide hydrogels in regenerative medicine, current fabrication methods fail to produce mechanically stable scaffolds using only hydrogels. The recently developed hybrid extrusion-based bioprinting process promises to resolve these current issues by facilitating the simultaneous printing of stiff thermoplastic polymers and softer hydrogels at different temperatures. Using layer-by-layer deposition, mechanically advantageous scaffolds can be produced by integrating the softer hydrogel matrix into a stiffer synthetic framework. This work demonstrates the fabrication of hybrid hydrogel-thermoplastic polymer scaffolds with tunable structural and chemical properties for applications in tissue engineering and regenerative medicine. Through an alternating deposition of polycaprolactone and alginate/carboxymethylcellulose gel strands, scaffolds with the desired architecture (e.g., filament thickness, pore size, macro-/microporosity), and rheological characteristics (e.g., swelling capacity, degradation rate, and wettability) were prepared. The hybrid fabrication approach allows the fine-tuning of wettability (approx. 50–75°), swelling (approx. 0–20× increased mass), degradability (approx. 2–30+ days), and mechanical strength (approx. 0.2–11 MPa) in the range between pure hydrogels and pure thermoplastic polymers, while providing a gradient of surface properties and good biocompatibility. The controlled degradability and permeability of the hydrogel component may also enable controlled drug delivery. Our work shows that the novel hybrid hydrogel-thermoplastic scaffolds with adjustable characteristics have immense potential for tissue engineering and can serve as templates for developing novel wound dressings.