Your assumption is correct. In the example "Después salgo", the word "después" is functioning as an adverb, indicating the timing of the action, and it is not used with the preposition "de". When "después" is used as an adverb, it can be followed directly by a verb in its conjugated form, as seen in your example with "salgo" (I leave).

However, if "después" were followed by the preposition "de", it would be followed by an infinitive verb. For example:

So, in the context of your sentence, "Después salgo" is a correct usage, and "después" is acting as an adverb.

If you have more questions or if there's anything else you'd like to clarify, feel free to ask!