AC units in humid climates need thicker coils to remove the latent (moisture) from the air. Restaurants, auditoriums, churches and such with high occupancies have similar problem even here in dry So. Cal.
You should check with an AC professional to see if it is possible to put a dehumidifier in the return air for your AC. This could reduce the latent load on your existing AC unit with less expense.
The best solution would be verify the capacity of your existing AC unit (condenser [outside] and evaporator [inside]) meets both the temperature and humidity requirements you want. Most designs are only for nominal environmental peaks, not extremes.
Secondary systems are sometimes a cost effective way to handle extremes without impacting the performance on nominal peak usage.
