Given the expense ("Why build one when you can build two for twice the price?"), loss of life and controversy, only for the pod to seemingly just fall through, there wasn't enough political willpower to fund a second attempt -- the Congressional hearings led by Kitz was an attempt to scapegoat the immense failure on Hadden as an elaborate hoax.
Conceivably there is a number of people (scientists, technicians, etc.) who are aware of the 18-hour recording discrepancy, but surely there are many (Kitz probably among them) who would want to cover it up or pass it off publicly as a crack-pot conspiracy theory. Given that, and the government's unwillingness to proceed further, it's probable no private entity would be willing buy out and fund such a PR disaster either.
Finally, while there's also nothing in the movie that says it can only be used once, it also doesn't say it can be used more than once. It's possible the machine was designed for a single trip only. However, that wouldn't stop a third (or more) machine from being built. The plans didn't cease to exist, provided someone was willing to pony up an exorbitant amount of cash.