

In this case it wouldn’t change anything. You would still need to ask the company. They might make you wait 30, 60 or 90 days or they might not.
Although, I didn’t ask where you are, and in this case it is relevant, as my answer is based on employment practices in the USA.
And no, open enrollment wouldn’t be relevant in this case, at least in the US. Changing employment status is generally considered a qualifying life event.