Yes, additional icons earned from previous Highlight of the Day bonuses do count as icons for set completion in bracketed Highlight of the Day bonuses.

Explanation

• On page 15, it states:
  "Some 'Highlight of the Day' bonuses, if their requirements have been met, give players icons that grant additional movement of a player’s Experience Track and Mood Track tokens. These earned icons should be treated and counted the same as the icons previously collected on the upper sections of the cards when scoring future days in the itinerary."
• This means that any icons gained from earlier Highlight of the Day bonuses are added to your total icon count for the purposes of scoring later bonuses, including those that require complete sets (brackets).

Example

• If a player has previously earned a Temples/Shrines icon from a Highlight of the Day bonus, that icon is included when counting complete sets for a later bracketed bonus that requires Temples/Shrines and Nature/Gardens icons.

Source: page 15.

Source: Base Rules (pp. 10, 15, 6, 7)