Given natural limitations on the length DNA sequences, designing phylogenetic reconstruction methods which are reliable under limited information is a crucial endeavor. There have been two approaches to this problem: reconstructing partial but reliable information about the tree (\cite{Mo07, DMR08,DHJ06,GMS08}), and reaching "deeper" in the tree through reconstruction of ancestral sequences. In the latter category, \cite{DMR06} settled an important conjecture of M.Steel, showing that, under the CFN model of evolution, all trees on $n$ leaves with edge lengths bounded by the Ising model phase transition can be recovered with high probability from genomes of length $O(\log n)$ with a polynomial time algorithm. Their methods had a running time of $O(n^{10})$. Here we enhance our methods from \cite{DHJ06} with the learning of ancestral sequences and provide an algorithm for reconstructing a sub-forest of the tree which is reliable given available data, without requiring a-priori known bounds on the edge lengths of the tree. Our methods are based on an intuitive minimum spanning tree approach and run in $O(n^3)$ time. For the case of full reconstruction of trees with edges under the phase transition, we maintain the same sequence length requirements as \cite{DMR06}, despite the considerably faster running time.