It sounds like a WILD too me.

Given you woke up tired, you woke up while your body immediately wanted more sleep. Apparently, the type of sleep your body wanted was REM sleep. It's easiest to have WILDs in this situation.

I'm not sure how to purposely replicate this situation of waking up tired without using alarms. Some use alarms to wake up tired using a technique called CANWILD (Custom Alarm Noise WILD) https://www.dreamviews.com/wake-init...al-%2A%2A.html I personally don't like waking up to alarms but it might be something you could try.

I like the WILD/DEILD technique on obe4u.com. https://obe4u.com/how-to-lucid-dream/ It's simple and fast and uses natural awakenings although it doesn't always work when waking up alert.

Sometimes I can WILD despite waking up alert I just imagine a dream scene or movement continuously while I fall asleep again, although this can require quite a bit self control, especially if I don't fall asleep quickly.

When waking up alert, one possible alternative is to wait until tired and then imagine a dream scene or movement continuously. There's a thread somewhere for this as well but I can't find it at the moment.

Hope this helps!