Is there anyway I can make the model thicker without having to redo it from the start?
Try to get front, back, top, bottom, left, and right reference images of things and making them background images with
N panel N >
Background Images. Make a 2D mesh for each reference and then try to combine them together to make a 3D mesh. I cannot tell you how exactly because it will vary with each model, but I can tell you that you can do CTRLJ to join the meshes and try to combine the appropriate vertices, and then fill in the faces.
I did this for a hawk's beak recently. What I did was that I made the side reference mesh, duplicated and made them 45 degrees apart (the front reference showed this), joined and connected them and made faces, applied a subsurf modifier, and it looked like an actual beak. So that tells you that this technique works. There are other methods, though!
Highlight the faces in edit mode and use 'E' to extrude the faces. Though this will only give you an extruded model like play-doh. If you're trying to recreate the model in 3D properly, you're going to have to learn polygonal modeling.