When exactly can the target make an Int check to end Detect Thoughts?

The spell description for Detect Thoughts in D&D 5e says that after you have done a deep probe “unless you shift your attention to another creature’s thoughts, the creature can use its action on its turn to make an Intelligence check” to end the spell. I assume they will pretty much always be able to do this at least once, since you have just used your action to probe their mind and won’t be able to change targets until your turn rolls around again, after their next turn.

My question: does searching for thoughts, the other action you can take with this spell, relevantly count as “shift[ing] your attention”? That is, if I probe a creature’s thoughts, and then on my next turn take the “search for thoughts” action with the spell, can that creature still contest the spell with an Int check afterwards?