In Greek, it's not uncommon to add a preposition to the beginning of a verb. Ballō means "throw," but ekballō means "throw out." Echō is "have," but katechō is "own." In Mark 1:36 we find the term katadiokō. Diokō means "pursue/search/hunt," but the kata intensifies the meaning: "hunt down." I think katadiokō is a fairly strong term in Mark. It probably means than "followed after him" (KJV) or "searched for him" (CSB, ESV, LSB) or even "pursued him" (RSV). Here's how we translated the verse in the ISV:
"Simon and his companions searched frantically for him."
In my Greek classes I'm always reminding my students of the importance of careful observation when exegeting a text. This is when we ask and answer the question: "What do I see?" When the palmist prayed, "Open my eyes that I may behold wonderful things from your law" (Psalm 119:18), he was praying for the power of observation. What makes one Greek student better than another? They can see more. That's all. The same information is available to both of them in the text. The big difference is what either one of them can observe.
Friend, observation is one of the most useful skills you can acquire as a student of God's word. There's a vast difference between merely seeing and carefully observing!