Yes, a player can defeat the same generic Wanted Bounty on both sides in the same action phase, even if they gain additional Attack after the first defeat, as long as they have enough Attack for each defeat.

This is supported by the following points from the rulebook:

• "You are allowed to defeat each side of a Wanted Bounty in one turn—example: you can spend 5 while to defeat Talbot’s Gang twice."
• "You must have enough Attack (generated by cards you’ve played or by converting into ) to defeat the Bounty. The amount of Attack a card generates is found in the upper left corner of cards that you play from your hand."
• "Whenever one of these Bounties are defeated, immediately collect your rewards and flip the card over. Wanted Bounties are always available to defeat and are never discarded."

There is no rule stating that all Attack must be spent at once or that you cannot generate more Attack between defeats. As long as you have enough Attack for each defeat during your action phase, you may defeat both sides of a Wanted Bounty.

Source: page 21.

Source: Core Rulebook (pp. 10, 14, 21, 22, 8)