A cover letter for bank jobs can be anything between 250 to 400 words and generally consists of 3 main sections. Choose a few phrases in the opening paragraph that let your reader know what to expect in the body of the letter.Find out how to capture your skills and flair for customer service all in a few paragraphs with our bank teller cover letter examples. Application Process - Starting Out. How do I apply for a job? Allpurpose cover letters are helpful for applying to several positions or to bring to job fairs. We regret to inform you that the job you were interested in has been closed. Application cover letter for the position of bank teller. Fill out the application. If the employer is well- known, the organization may be placed before the job title.