Rule 1.3c covers this and goes on to clarify what happens in general when a player breaches a rule and then breaches another rule before the next stroke.
As of January 1, 2025, in competitions where the Model Local Rule is in effect, visible cracks to the face and clubhead now also constitute “broken or the damage significant” enough to allow the club to be removed (providing it was not caused by player abuse).
In most instances, the clock does stop and you get to resume your search with whatever time is remaining from when you halted the search. But in other cases, it’s not quite that simple.
Rule 4.3 allows you to carry things such as an extra driver head or shaft in your bag. Grips are OK, too. What you can’t do is use any of that stuff during the round.
While most people see the rules as limiting what golfers can do on the course, a broader understanding of golf’s dos and don’ts allows players to take advantage of opportunities the rules also present – and keep your scorecard from blowing up in the process.