In this work, we report for the first time, growth of secondary carbon nanotubes (CNTs) throughout a three-dimensional assembly of CNTs. The assembly of nanotubes was in the form of aligned CNT/carbon (ACNT/C) foams. These low-density CNT foams were conformally coated with an alumina buffer layer using atomic layer deposition. Chemical vapor deposition was further used to grow new CNTs. The CNT foam's extremely high porosity allowed for growth of secondary CNTs inside the bulk of the foams. Due to the heavy growth of new nanotubes, density of the foams increased more than 2.5 times. Secondary nanotubes had the same graphitic quality as the primary CNTs. Microscopy and chemical analysis revealed that the thickness of the buffer layer affected the diameter, nucleation density as well as growth uniformity across the thickness of the foams. The effects of secondary nanotubes on the compressive mechanical properties of the foams was also investigated.