The H-1B visa is employer specific. Thus, you will have to obtain an H-1B for your applicant even though s/he may already be in H-1B status with another employer. These cases are considered "new" employment and will require the $500 anit-fraud fee.
Under the "portability" provisions of the law, the employee may begin work with K-State after the H-1B is filed with USCIS and the receipt notice is received by International Student and Scholar Services.
Can we apply for an H-1B for a part-time position?
An H-1B visa can be applied for a part-time position. The Department of Labor (DOL) regulations require employers to keep records of "hours worked each day and each week" for all part-time H-1B employees, regardless of whether or not the part-time employee is paid a fixed salary, and regardless of whether or not the employer currently keeps such hourly records for its other part-time salaried employees. It is the department's responsibility to document these hours and have them available if requested in a Department of Homeland Security (DHS) or DOL audit. A Part Time Certification form is required for the H-1B petition if the position is less than full-time.