Space Shuttle Columbia entered a period of peak heating at 46 mi (74 km) altitude. Before the contact got broken, the temperature on Columbia's wing leading-edges had sunk already (and the heat shield damage was on a wing). Why did the disaster happen despite being past the hardest phase of reentry? During this phase the shuttle broke up more and more, so was this due to the additional damage during the reentry? It eventually disintegrated around 39 mi (63 km) altitude.
The story of the destruction of the orbiter during entry is told in extreme detail in Section 4 of Appendix D.7 of the Columbia Accident Investigation Board report Volume 2.1
The very short version of the story is that a small plume of plasma entered the wing through the breach in the wing's leading edge thermal protection system, and it took a while for this plume to cause sufficient damage for the wing to suffer structural failure.
The wikipedia link in the question states that at Entry Interface (EI) + 404 seconds, "Columbia entered a 10-minute period of peak heating, during which the thermal stresses were at their maximum." Let's take a look at what that means.
This graph shows the heating rate on the wing leading edge panel.
There doesn't appear to be anything special about EI+404 based on this graph. The heating rate continues to increase until ~ EI+620, then plateaus until ~ET+750.
The document records the progression of the damage as the plume ate away at the wing insides:
The centerline of the plume contained enough energy to begin melting the exterior of the wheel well wall by approximately 8:54:00 EST (EI + 594 sec.).
By 8:56:16 EST (EI + 727 sec.), hot gas had penetrated the wheel well wall
... 8:57:19 EST (EI + 790 sec.) .... This is an indication of extreme heating of both the left outboard tire and the surrounding instrumentation. The tire has significant thermal mass and substantial heating would be required to produce the slight temperature rise. The tire has significant thermal mass and substantial heating would be required to produce the slight temperature rise. By 8:58:56 EST (EI + 887 sec.), all left main gear inboard and outboard tire pressure and wheel temperature measurements were lost indicating a rapid progression of damage or wire burning inside of the wheel well.
The side-slip angle changed sign at 8:59:36 EST (EI + 927 sec.) indicating that vehicle loss of control was imminent..
main vehicle aerodynamic break-up occurred at 9:00:18 EST (EI + 969 sec.),
So to directly address the question: Why did Columbia disintegrate lower into the atmosphere, not during its peak heat phase?:
It was still within the peak heat phase, even by the wiki definition (EI+404 seconds + 600 seconds = 1,004 seconds). It broke up at EI+969 seconds.
1 At least for me the pdfs on this page do not work when clicked. I have to download them first to a file, then open them. If I just click them, I get a scrambled ASCII file.