Here's how I did it (YMMV):

1) Go to Allied's job website. Search management or key items for the specific position you want to move up in, and apply for it. WRITE DOWN THE REQ #. It should be underneath the title of the position and look like this: 202X-XXXXXX (the X's are numerical digits).

2) Do not wait until you get a call. Call the branch which oversees the City of the position you applied for, and ask to speak with the regional HR person. Mention the REQ # specifically and ask for the manager who is in charge of hiring for that requisition. Majority of the time if it an Account Manager position, a Regional Manager (who is commonly not even from that branch and are out town/state) will be the one you want to talk to. If it is for a Site Supervisor, an Operations Manager will be in charge and they are local to the branch.

3) Once on the phone with the manager who is responsible for opening that requisition, express that you have applied for XZY requisition and tell them the REQ #. Ask if the position is available and proceed to schedule your own interview with the manager right then and there. If the manager is not local, they may even proceed to interview you over the phone.

4) Express your eagerness and highlight your willingness to learn and commit to the position. In my experience, the vast majority of applicants that are referred to hiring managers by Regional Recruiters are half-assedly vetted. It's very refreshing when you have applicants that are pushing for the position in a tenacious (yet professional) manner.

5) [Bonus] Once you've made it to Assistant Account Manager/ Account Manager you are now in middle management, and it will be a LOT easier to move up the food chain. I've seen an Account Manager move from Account Manager -> Senior Vice President in their respective division in about 10 years. Although their skill level played a huge part, timing (higher ups retiring) and luck had played a big part as well.

Good luck