In this work, we introduce new construction methods for self-dual codes using a Baumert-Hall array. We apply the constructions over the alphabets F_2 and F_4 + uF_4 and combine them with extension theorems and neighboring constructions. As a result, we construct 46 new extremal binary self-dual codes of length 68, 26 new best known Type II codes of length 72 and 8 new extremal Type II codes of length 80 that lead to new 3-(80,16,665) designs. Among the new codes of length 68 are the examples of codes with the rare γ= 5 parameter in W68;2. All these new codes are tabulated in the paper.