I can. Second Wind is a clear victory for your playstyle over mine, and I don't see them offering any alternatives for controversial stuff that exists in Basic D&D, because it's supposed to be part of the assumed core of the game and you can't get any more core than a level 1 basic fighter.
Put it this way, if there is an alternate Second Wind in the DMG, it would be like them admitting they deliberately chose (which they did), a controversial mechanic for the most basic class of the game which has caused endless debate since the 4e era, as a giveaway to fans of that game to get them on board and buy the books, over the objections of others who have very good reasons to, because HP aren't stamina in D&D. D&D doesn't model stamina in a round by round fashion, never has and never will. It's too boring to even contemplate how tired someone is after 6 seconds of combat or why, if they are incredibly tired after such a short time, are they out adventuring and not in a retirement home for ageing adventurers, or why aren't they taking a -1 penalty to hit after each round of combat. Or why you can swing your sword arm all day but it's only when the giant's hammer hits you squarely on the head do you get tired (because you can now use Second Wind to restore that lost "stamina")