Premise: I work for a big US enterprise company, I travel a lot and I carry around my work laptop, seldom my personal machine.
Contract-wise, other than a non-competition clause, it is stated that anything I do using company assets belongs to the company.
Disregarding the consequences that would affect me, what kind of claim would they get on the project itself, if the company came to find out about my breach of contract? Is it even breaching contract?
Talk with your manager some more, see how much s/he can help you fix this with the company. This might be getting the OK for open source work (future, and retroactive), or it might be only getting you a pass for a past misunderstanding.
If manager decides to make a case for the company supporting/tolerating open source contributions, s/he might sell it as a way to improve employee morale, enhance the reputation of the company for recruiting, improve customer goodwill/reputation for open source savvy, get free open source labor from others on software the company can use internally, or something like that.
You might find it reassuring to consult your own lawyer. Your own lawyer might have a very different perspective on some key things (e.g., reasonable personal accommodations while traveling on business), but hopefully it won't be necessary to debate the matters. If someone asks you to sign something, or your gut says you've been targeted for termination and/or legal action, consult your own lawyer.
Regardless of how that turns out, you might also need to give the leadership of the open source project(s) a heads-up on the status of your contributions, in a timely manner. They might need consider when they need to do anything now, to minimize headaches for anyone.