Are there any feats, classes that can be multi-classed into, or any other mechanic, that would give someone expertise in smith's tools (i.e. double their proficiency bonus), that don't require one to start as a certain class? I was trying and failing to find any.
The only way I can find relies on Unearthed Arcana material, and requires that you not have proficiency with smith's tools, so it may or may not be useful to you.
The Artificer, from Unearthed Arcana, gets this feature:
Starting at 2nd level, your proficiency bonus is doubled for any ability check you make that uses any of the tool proficiencies you gain from this class.
Since Unearthed Arcana isn't designed to be multiclassed, we don't know whether you would gain tool proficiencies when multiclassing into it. Luckily, we can choose the Gunsmith archetype, which gives us this feature:
When you choose this specialization at 1st level, you gain proficiency with smith’s tools, and you learn the mending cantrip.
So assuming your DM allows you to use Unearthed Arcana material, and allows multiclassing with it despite it not being designed for multiclassing, 2 levels in Artificer will get you expertise in smith's tools.