It is usually best to pinpoint your ultimate goal and work backwards from there. If your desire is to become a GM of a large-scale operation, then you will need to gain the necessary experience within the industry. You may find the most success based on the size of the hotel and the organizational structure. For example, larger hotels utilize a Guest Services Manager which is an outstanding way to gain supervisory/managerial experience and I would imagine you would be competitive based on your work experience. From here, you can work towards department manager positions and ultimately move into the top spot at the property. Worst case scenario, you can apply for departmental "leader" positions that are great ways to prove your abilities and go from there. It may be more difficult to find leadership roles at smaller properties since there are only a small amount of managers and supervisors.
In my experience as a hiring manager, it is typically rare for an individual to jump right into the industry into any managerial position.
Best of luck.