Yes, the monster immediately enters play when revealed, even if the round is not yet over.

This is supported by the following from page 21:

• "After defeating a Most Wanted Bounty, take the Bounty card and place it in your score area. Immediately flip over and reveal the next bounty in that deck."
• "Setting up a Monster Figure: When you flip over a Monster Card, grab and place its figure on the Last Seen Location. The figure will move around the map at the end of each round."

This means:

• As soon as a Monster card is revealed (by flipping the next card after defeating a Most Wanted Bounty), you immediately place its figure on the board at the Last Seen Location.
• The monster does not wait until the end of the round to enter play; it is placed as soon as it is revealed.

Source: page 21.

Source: Core Rulebook (pp. 10, 11, 21, 23, 24)