How small can the error be in LWE?
Per Mark's suggestion, I looked into the &quot;hidden&quot; tests in lwe-estimator and read a few papers. I summarized my findings as an answer here:

Arora-Ge attack, and the improved version using Grobner bases, work better when $q$ is small, but it starts to be impractical once the number of samples $n$ and $q$ is very large, e.g., $q=2^{64}$. This seems to be due to the time needed for linearization.

Another hidden test, MITM (Section 5.1 in https://eprint.iacr.org/2015/046.pdf), can be useful here---MITM considers $\alpha q$, and it is important that $\alpha$ cannot be way too small. Reflected in lwe-estimator, this would disallow one to use $\sigma\approx0.001$.


Still, more discussion is welcome. For my own use case, it seems that $\sigma=1$ or $\sigma\approx0.8$ may be generally okay since my $q$ is large.