My understanding is that one can avoid a stopover by taking less than a 4 hr connection OR taking the next available flight (which will often be more then 4 hr in cases where the next flight is the next day).

Having said that, when you search AA it does not show you all possible connections; it shows you the "best" ones (ranked by however they decide to rank them). If you can price it out on GF it may give you a direct link to book on AA. If not, you can use matrix and either use bookwithmatrix, or use the fare construction at the bottom and either try to get an AA agent to replicate it, or just send it to a travel agent.