‘Most’ is used with an adjective or adverb to form a superlative. It is whether this superlative is in reference to a noun or a verb that determines use of the article. When used with the verb ‘to be’, it helps to look at the superlative as referring to either an object (noun) or an action (verb).
Here is a sentence where the superlative refers to the object (noun):
He’s the most approachable on staff.
This sentence uses ‘the most’ because the superlative refers to the object (an approachable person on staff).
In your sample sentence, however, the superlative refers to the action:
He’s most approachable first thing in the morning.
What’s amplified here is the action of being approachable at a particular time. This person may be extremely unapproachable in general.
In the first sentence of user307254’s answer, the superlative refers to the object (a challenging job he has had), which necessitates ‘the most’.
The second sentence refers to the action (being persuasive) - in this case you would leave off the article.