The duties performed in your job(s) are going to define your position.
To be PC and not wanting to “hurt” anyone’s feelings, job titles took on the “importance factor,” such as, homemaker=domestic engineer, janitor=environmental engineer, etc. Everyone has a technician or engineer moniker attached to their job.
Going from your current job with the Admin title to the larger company with the help desk title will not hurt you in any way. When interviewing in the future, an employer is going to look at your accomplishments and determine that both jobs were close in their daily tasks.
My suggestion would be to take the job for more experience. Processes and interaction with more people will help you with your “soft skills” which are very important for success.
As for the long-term, it depends what you intend to do. Large corporations employ specialists in a field instead of generalists, and you will eventually need to specialize in an area. This may be more possible at the new company that is a little larger, with 6 co-workers, which may let you become the “technology X” expert there, and use that experience as a catalyst to a large corporation.
If you want to work for small to medium companies, then it won’t matter that you went from “admin” to “help desk” because the companies that comprise this segment usually employ generalists because they don’t have the money to spend for specialists and they don’t want 500 IT employees.
If your intention is to go into IT management, then moving to the new company would benefit you by working with other employees and learning how that organization works. Take with you all of the good AND bad that you can from each company you work for and use the knowledge to help you excel in your career. If you want to work in management for a large corporation, you would most likely have to start out as a specialist and then over the years (and it can take YEARS) move up into a departmental manager, then move to a area manager, to division manager, and then move into the vice presidential positions. Remember though, these moves are extremely political and generally take many years to reach; though some do move faster than others, for whatever reasons (read, political).
Decide what/where you want to go and that will help you in deciding if the new job is what you feel is best for you.