Having somebody else make it
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Contract manufacturing exists in soap and it sits differently than in the other crafts, because the product is simple and the brand is the person.
Less common here, and it happens
Bar soap is easy to make and hard to make a lot of, so the reason to outsource is capacity rather than capability.
The usual routes are a contract soap maker producing to your recipe, or a private label supplier whose base you brand.
Those are very different things and are worth not confusing.
What you are handing over
To a contract maker: your recipe, your process and your specification, and you keep the formula.
To a private label supplier: nothing, because it is their formula and you are buying their bar with your band on it.
Which decides whether you can ever move the production, and whether the product is actually yours.
What it solves
Cure space, which is the constraint, and it is now their problem.
Capacity for a large wholesale order you could not otherwise promise.
Consistency, if they are good, since a production operation holds a process better than a person in a kitchen.
And your time, which moves from making to selling.
What it costs you
Margin, obviously.
Minimum order quantities, which for soap are large and paid up front.
And the story, which in this craft is a substantial part of the value. A customer buying handmade soap at a market is buying that it was made by the person at the table.
So the honest question is whether your customers are buying the bar or buying the maker, and for most small soap businesses it is the maker.