A serie of the Tl-2201 cuprate with Tc ranging from ∼ 40 K to 92 K with tetragonal as well as orthorhombic symmetry has been prepared and their differential magnetic susceptibility χ'H measured under static field up to 2 Tesla. Independently of the criteria defining the irreversibility onset, we show that the extension of the reversible regime in the (H,T/Tc) diagram increases with Tc. Thus, contrary to what has been observed for YBa2Cu3O6+x where the reversibility domain was enlarged with the Tc decreasing, here for low Tc the reversibility domain is more limited and shifted toward Hc2(T) line. These results are discussed taking into consideration the evolution of λ and the possible evolution of ξ with Tc in this Cu-O monolayer overdoped HTS.