Given two graphs
, a graph
-free if it contains no subgraph isomorphic to
. We continue a recent study into the clique-width of
-free graphs and present three new classes of
-free graphs that have bounded clique-width. We also show the implications of our results for the computational complexity of the
problem restricted to
-free graphs. The three new graph classes have in common that one of their two forbidden induced subgraphs is the diamond (the graph obtained from a clique on four vertices by deleting one edge). To prove boundedness of their clique-width we develop a technique based on bounding clique covering number in combination with reduction to subclasses of perfect graphs.