How does our body produce new antibodies?
In case of T-cells and Abs our organism first generates a great diversity of T- and B-cell receptor variants so that any new foreign antigen is likely to be recognized by some of them. When a B-cell encounters its antigen it begins proliferating and changes it phenotype. Additionally, B-cell (but not T-cell) receptor starts to mutate (so-called somatic hypermutation) generating new B-cell receptor variants that differ a bit from the original one. This process acts like evolution, with B-cell carrying a receptor with higher affinity being more likely to proliferate. So the specificity of antibodies improves indeed, this is called affinity maturation.