Preterm infants develop differently than those born at term and are at higher risk of brain pathology. Thus, an understanding of their development is of particular importance. Diffusion tensor imaging (DTI) of preterm infants offers a window into brain development at a very early age, an age at which that development is not yet fully understood. Recent works have used DTI to analyze structural connectome of the brain scans using network analysis. These studies have shown that, even from infancy, the brain exhibits small-world properties. Here we examine a cohort of 47 normal preterm neonates (i.e., without brain injury and with normal neurodevelopment at 18 months of age) scanned between 27 and 45 weeks post-menstrual age to further the understanding of how the structural connectome develops. We use fullbrain tractography to find white matter tracts between the 90 cortical and sub-cortical regions defined in the University of North Carolina Chapel Hill neonatal atlas. We then analyze the resulting connectomes and explore the differences between weighting edges by tract count versus fractional anisotropy. We observe that the brain networks in preterm infants, much like infants born at term, show high efficiency and clustering measures across a range of network scales. Further, the development of many individual region-pair connections, particularly in the frontal and occipital lobes, is significantly correlated with age. Finally, we observe that the preterm infant connectome remains highly efficient yet becomes more clustered across this age range, leading to a significant increase in its small-world structure.