I asked to delay my start date at my new job. They approved, but I'm wondering if I should give them some additional context to clear the air.

I was originally scheduled to start my new job on the 26th, but earlier this week I asked to push it back to the 3rd. They approved, but with a gentle "warning" that they typically prefer additional notice for things like that (I.e. vacation time). I definitely understand where they're coming from; since they are off all of next week, I effectively gave them only a week of notice.

That said, part of the reason I asked to move the start date was because I was originally planning to be on vacation all of next week myself. Unfortunately, because the background check didn't clear until last Wednesday, I didn't put in my notice with my current job until then and now have to finish up my two weeks next Wednesday.

I'm wondering if I should tell them this. I don't think what I did is necessarily a red flag or anything for them, but I don't want to start off on the wrong foot in any way, and I don't know if this context would help at all. Thoughts?

EDIT: I don't really think it's all that relevant, but I guess I should say that the other reason I asked for the delay was to buy some extra time to consider another potential job offer coming up. I obviously won't be telling the first job this.

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