I don't honestly know. I didn't work on the orbiter - I did the Hubble Space Telescope servicing mission payloads. I don't know what techniques they've figured out after the accident, but I'm not sure there was a reliable way to go out and look at the time. The shuttle's arm is only so long, and even if you could put an astronaut in postion to look, the EVA time required would likely be prohibitive. You'd need some way to inspect the underside of the orbiter from a satellite, earth, or some other vantage point. I'm sure google can tell us what they'd do today to help mitigate this kind of failure.

IIRC, two ways to inspect the orbiter's underside were developed - an inspection boom that could be attached to the end of the arm equipped with sensors (visual and laser), and a maneuver procedure before docking with the ISS - the orbiter would make a full rotation allowing it to be photographed from the ISS. During one of the first post-Columbia flights, a problem - a spacer which protruded from its proper position between two tiles was removed during an EVA. This rotation maneuver was the most shocking to me - as it showed it never occur to anyone just to rotate the shuttle so it could be inspected with a pair of binoculars. Such an inspection could be conducted as early as STS-63.

In a sense, every shuttle was an X-plane - it was as much a research vehicle as a commercial transport to LEO. One builds shuttles to learn how to better build shuttles and, in order to do that, learn as much as possible.

BTW, I envy you (in a good way, of course) more than a little. Working for NASA is a really cool thing.

Going to extreme lengths they could have examined the heat shield by repurposing spy satellites. But there was very little they could have done about it anyway.

My understanding is that the effort wasn't extended with the understanding that the crew were doomed regardless of what information was gathered.

My view is that even if the crew were doomed, gathering additional information in advance of reentry would have allowed for a better understanding of circumstances, better post-disaster modeling of what went wrong and how the orbiter failed (both in the launch-time foam strike, and in the reentry heat-shield penetration and structural failure).

Whether or not to inform the crew is yet another decision. Astronauts are aware that theirs is a highly risky venture, though with a low sample size, the specific odds are somewhat uncertain, though on the order of 4:100 per human space flight. If you're going to go on a space mission, you'd better be prepared to die.

If NASA refrained from assessing strike damage on those grounds, I feel a grave error was committed.

An engineer started arranging spysat imaging through "back channels" but was struck down by management - he/she didn't have the guts to escalate it into an official thing.