First of all, please mind your spelling:
- The noun describing the time of the day or the future is capitalized: Der Morgen / das Morgen.
- The adverb describing that something is happening the next day is written in lower case: morgen
Now obviously something can happen in the morning of the next day and morgen technically stretches until midnight of the following day. In such cases you add a qualifier and say
- morgen früh (in the morning) or
- morgen Vormittag (before noon)
Remember that a most people “start” their new day in the morning after a night’s rest. (The concept of a day starting at midnight is a relatively modern one and does not reflect how we humans instinctively perceive our time.) So der Morgen (the new day) starts morgens (in the morning) - really the same concept.
And just for completeness:
Ein Morgen used to be the area a single worker could till in one Morgen or Vormittag. It’s an unit of measurement which derived its name from the time span.