Yes. You can apply for positions as a full charge bookkeeper after completing the Bookkeeping Program.
Realize that no matter what education you get, you will not use 100% of the education. It will all depend upon the business. As a simple example, an accountant learns cost accounting and governmental accounting. If that person works for a manufacturer, they will never use governmental and vice versa. Even within cost accounting, that person learns process cost accounting and job cost accounting. Depending upon where they work, they will only use one half of what they learned in the one subject Cost Accounting.
So, while you have gained additional knowledge, if you go to work for a construction company, the knowledge you will use will be different from working for a car dealership. Both will use A/P and the concepts remain the same, however, as a simple example, in a car dealership you may never use chain discounts but use interest calculations all of the time whereas in the construction company it could be the opposite.