In the first part of the paper, we adapt techniques used in previous works and heavily expand and generalize them. In particular, in [6] a temporal spanner in cliques is computed by a reduction from a clique to a bi-clique. We deepen this connection by showing that the existence of a linear spanner in cliques and bi-cliques is equivalent. Furthermore, in the reduction of Casteigts, Peters, and Schoeters [6], the authors exploit the property that the set containing the earliest edge of each vertex is a perfect matching of the bi-clique and that this also holds for the set of latest edges. We show that this property can also be assumed when searching for bi-spanners in bi-cliques directly. This yields a simpler and more intuitive proof, that temporal cliques admit temporal spanners of size O(n log n). Moreover, we reconsider the notion of a pivot-vertex [6, 7]. This is a vertex u that can be reached by every vertex in the graph until some time step t, and then u can reach every other vertex in the graph. We transfer this notion to bi-cliques, extend it to pivot-edges, and we strengthen it significantly to find small spanners. If a bi-clique contains a pivot-edge, we can iteratively reduce the size of the graph, while maintaining a linear-size temporal spanner, until the instance is solved, or the instance no longer has a pivot-edge. This shows that a large class of graphs admits linear-size temporal spanners. In the second part of the paper, we investigate temporal cliques that are not edge-pivot graphs. We found two such graph classes, called shifted matching graphs and product graphs, and for both we provide novel techniques that allow us to show that linear-size spanners always exist. Thus we extend the class of temporal cliques that admit linear-size temporal spanners even further. It is open if our classes cover every temporal clique. However, finding other graphs that do not belong to our graph classes seems to be challenging.