Peg solitaire is a classic game where players aim to remove pegs from a board via a jump move. In the context of graph theory, this game introduces an interesting challenge: for a given graph G, how many edges must be added to make it solvable in peg solitaire? This quantity is known as the minimal solvability number, ms(G). We study the ms(G) specifically for caterpillar graphs, identifying types with known minimal solvability numbers and exploring families, such as caterpillars of length 3, that remain unsolved. For this family, we provide upper bounds on the ms(G) and identify cases where ms(G) = 1.