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The computation of the maximum toughness among graphs with \( n \) vertices and \( m \) edges is considered for \( \lceil{5n}/{2}\rceil \leq m < 3n \). We show that there are only finitely many cases in which the toughness value \( \frac{5}{2} \) cannot be achieved. This is in stark contrast with the known result that there is a \( \frac{3}{2} \)-tough graph on \( n \) vertices and \( \lceil{3n}/{2}\rceil \) edges if and only if \( n \equiv 0, 5 \pmod{6} \). However, constructions related to those used in the cubic case are also employed here. Our constructions additionally provide an infinite family of graphs that are supertough and not \( K_{1,3} \)-free.