As long as you want all bit patterns to be Some float, there's not really one bit pattern you can chop out of somewhere (or, three, rather - both infinities, and NaN).

Taking, say, the 3 smallest subnormal numbers, or the three largest numbers, or whatever, would be extremely bad for allowing optimizations/algorithms/correctness checkers to reason about operations in the abstract.

You can put diagnostic information about the cause of the NaN in the bits. IEEE754 doesn’t mandate any format though, leaving it up to the implementation. You can check section 6 of IEEE754:2019 (which also talks about the possibility to use it to extend the standard to wider ranges, more infinities or other stuff)