Update 22 Dec 2013: This feature does not seem to be working in Sense 5 after I updated my HTC One X to Android 4.2.2.
While putting in contacts in my phone I generally put the Company name also. But to search by Company name you need to enable a setting which is not enabled by default in your People app of HTC One X.
Go to your People app and open the menu. Tap on "Settings".
From the options provided mark "Company" as selected. Save the settings.
And now you can search your contacts by Company name.