In this paper, for odd n and any k with gcd(n, k) = 1, new binary sequence families S^k of period 2ⁿ-1 are constructed. These families have maximum correlation $1+2^{n+3\\over 2}$, family size 2²ⁿ+2ⁿ+1 and maximum linear span $n(n+1)\\over 2$. The correlation distribution of S^k is completely determined as well. Compared with the modified Gold codes with the same family size, the proposed families have the same period and correlation properties, but larger linear span. As good candidates with low correlation and large family size, the new families contain the Gold sequences and the Gold-like sequences. Furthermore, S^k includes a subfamily $\\mathcal{S}^k_1$ which has the same period, correlation distribution, family size and linear span as the family So(2) recently constructed by Yu and Gong. In particular, when k=1, $\\mathcal{S}^k_1$ is exactly So(2).