If there is an open criminal case on a person, can the arresting deputy make any contact with the suspect in that case?

Contact is a word that encompasses verbal and non-verbal communication and physical touching. In executing an arrest, a police officer—as a general point of fact—must come into contact with the person being arrested.
For instance, a person placed under arrest may be subject to a physical search by the arresting officer (see Chimel v. California), which would require the officer to touch the person. More specifically, however, police must provide "Miranda warnings" to a person who has been arrested (see Miranda v. Arizona). This requires the arresting officer to clearly articulate, using verbal communication, several cornerstone constitutional rights to the arrested individual.
More generally, however, duly commissioned or deputized law enforcement officers have authority exceeding that of normal citizens, not less.
In cases of a criminal case where an arrest was previously effected, police can contact the suspect in that case in the normal course of investigation and subject to the requirements of law (e.g., a suspect can decline to answer most questions without the presence of an attorney). There is no statutory or constitutional barrier that prevents any specific officer from doing so.