A given API may have a bunch of overloads on an open
command but what's happening underneath is pretty much the same. Unless you switch to a completely different API or another db there is no way to "speed" up the process.
The only thing you can do is to mitigate the issue by moving the opening of the connection to some other part of the application's life-cycle where the user might not notice or mind a delay.
One trick that is widely used is to implement this slight-of-hand is to do your initialization while a splash screen is being displayed to the user. The splash screen might have some fancy graphics and text to "entertain" the end user. You will even see the splash screen updated with what the app is doing at that particular time. Not terribly useful but might be entertaining enough to the user to distract from the delay.
I did this once when users were complaining of an initialization step taking too long. After I published the change the end users swore that the app performed much better. It didn't really. I only manipulated perception.