It means "You're very welcome to contact me (I don't know if they mean email, mail, call or text).

Hi , it means exactly what it says. However it's often used as a way of ending a business or social exchange. If you do contact the person & they don't respond, you'll know they didn't really mean it.

it means 有空的时候多联系

It means: You are invited and welcome to email me or call me again, and we can continue, however I will not be contacting you. If you want to continue, you will need to contact me. Example: I am a teacher who is giving advice to a student. I answered the student's question, and the interaction is over. But I would welcome having this interaction again, but only if the student needs it. So I would tell the student "Please feel free to contact me" to warmly invite the student to ask another question, but to make it clear that I will not be contacting the student unless they ask. It is often used in business interactions to mean "I'm done with this topic, but we can discuss this more if you need to."

it means don`t feel shy, relax and contact me