Blast hits disappearing after changing -evalue
I am the author of the question / answer here Why does a very strong BLAST hit get lost when I change num_alignments, num_descriptions or max_target_seqs parameter?


  To my knowledge changing the evalue threshold should no change the
  values themselves, it should just change the reporting.


That is sadly not the case, though thats what I would say is typically expected.

Yours was the exact situation that lead me to question the implementation of NCBI blast. I got results with an evalue above a certain value with a relaxed threshold, but with a stricter threshold (still lower than the value of the results I got) those results disappeared.

From what I found all parameters imposing some threshold show this behavior and are applied in an early step of the program leading to prematurely removed results that later on might score higher than those thresholds. (Say your hit first has an evalue of 5: threshold 10 = ok and kept, threshold 1e-5 = gets removed, but that same hit gets improved further along in the program and then gets an evalue of 1e-6. Now this is only shown when you use an evalue threshold of 5 or higher, though 1e-6 is clearly also a result you would want to see when using a threshold of 3, 1, 0.01, ... etc.)

I was relatively shocked by this and the fact that its basically nowhere officially documented and generally a not known fact what influence these options have.

Sadly for now the stance of the developers is basically all is working as intended and the only feedback I got amounts to 


  The safest approach is to relax (increase) the threshold and deal with a larger result set.


aka Suck it up and deal with it.