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Finding a tag that introduced a submodule change

I have a repository with a single submodule in it. Since the parent
repository represents the code base for an actual product, I tag
release versions in the parent repository. I do not put tags in the
submodule since multiple other products may be using it there and I
wanted to avoid ambiguous tags.

Sometimes I run into a situation where I need to find out which
release of the product a submodule change was introduced in. This is
nontrivial, since there are no tags in the submodule itself. This is
one thing I tried:

1. Do a `git log` in the submodule to find the SHA1 representing the
change I want to check for
2. In the parent repository, do a git log with pickaxe to determine
when the submodule itself changed to the value of that SHA1.
3. Based on the result of #2, do a `git tag --contains` to see the
lowest-version tag that contains the SHA1, which will identify the
first release that introduced that change

However, I was not able to get past #2 because apparently there are
cases where when we move the submodule "forward", we skip over
commits, so the value of the submodule itself never was set to that

I'm at a loss here on how to easily do this. Can someone recommend a
way to do this? Obviously the easier the better, as I have to somehow
train my team how to do this on their own.

Thanks in advance.