Where did the rest of the elements come from?
By the "original periodic table" I assume you're talking about the "elements ocurring in measurable quantities in nature".--everything till Uranium sans Technetium and Promethium.

The method of preparation of these elements is pretty much the same -- bombardment. Just the projectile and substrate differ.For example, the (accidental) neutron bombardment of Molybdenum in old cyclotron parts led to the discovery of Technetium.

There are some other sources as well. Analyzing fission products can get you new elements--that's how Promethium was discovered. The process of fission creates smaller nuclei--but there is a shower of neutrons created as well (which can bombard everything in sight). Most of the discoveries have been due to the "shower of neutrinos", because generally all "smaller nuclei" are of already-known elements. Promethium was synthesized by the "formation of smaller nuclei", though.

Since fission occurs in nature (Usually it's just a little bit, though we sometimes have natural fission reactors), we actually have all the elements up till californium existing in minute quantities in nature (source).