Going out "for the day" implies spending the daytime (which in this case can be defined as the time one leaves home in the morning until the time one returns home in the evening) going somewhere (the beach, for example, or to visit friends) but would not include sleeping away from home.
You might go to Brighton for the day, but if you stayed out for twenty-four hours, returning the following morning having slept in a hotel that night, then you'd have gone out "overnight."
So you might say:
"We only intended to go to Brighton for the day, but it was so delightful we ended up staying overnight."
"For the day" will very rarely encompass an entire day, as the accepted answer suggests.
In your example, the writer is anxious about going on holiday (which involves several overnight stays) and is pointing out that there is no precedent in holidaying with Jan because they haven't "even" been out for the day, let alone stayed out overnight together.