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Phylogenetic placement, used widely in ecological analyses, seeks to add a new species to an existing tree. A deep learning approach was previously proposed to estimate the distance between query and backbone species by building a map from gene sequences to a high-dimensional space that preserves species tree distances. They then use a distance-based placement method to place the queries on that species tree. In this paper, we examine the appropriate geometry for faithfully representing tree distances while embedding gene sequences. Theory predicts that hyperbolic spaces should provide a drastic reduction in distance distortion compared to the conventional Euclidean space. Nevertheless, hyperbolic embedding imposes its own unique challenges related to arithmetic operations, exponentially-growing functions, and limited bit precision, and we address these challenges. Our results confirm that hyperbolic embeddings have substantially lower distance errors than Euclidean space. However, these better-estimated distances do not always lead to better phylogenetic placement. We then show that the deep learning framework can be used not just to place on a backbone tree but to update it to obtain a fully resolved tree. With our hyperbolic embedding framework, species trees can be updated remarkably accurately with only a handful of genes.