Yes, the phrase "then I just got out of bed" is correct and means that the speaker has recently awoken and left their bed. It can imply that they are still in the early stages of their day.
Here are a few ways to rephrase it:
1. "I just got out of bed."
2. "I just woke up and got out of bed."
3. "I’ve only recently gotten out of bed."
4. "I’ve just gotten out of bed."
5. "I just crawled out of bed."
All of these rephrasings convey a