One of the best things that you can do if you are in the market for a new HOA Management Company is to create a complete list of "needs and wants." The Board of Directors for the HOA may do this, or a subcommittee may be created, but it is important to know what it is you are looking for and be able to articulate this to the prospective companies that present their proposals.
A lot of HOAs are cost-sensitive. And if this is the case, you should consider what you "need" the management company to do vs. what you would like them to do. It would be better to have better quality service from a reputable, experienced company which did not do everything... than a company which did everything poorly!
When interviewing the Property Management Companies, ask them how they would have handled the situations for which brought you to the decision to look for a new company! This will give you an idea of the creativity and ability of the companies AND give you an idea of how reasonable your expectations are. Your requests may be easily met--or there may be a challenge for which you had not considered. Finding the right HOA Management company is really about a partnership, and they are just as responsible for telling you what is not possible as you are for asking for what is.
Like with much else, communication is the key. Ask for what you want, listen to what can be done, and set the expectations so that everyone is on board and your next HOA Management company can be the permanent management company.