Set iPhone volume to maximum when connecting to a Bluetooth speaker
Some Bluetooth speakers use the phone volume control to determine the speaker volume which is much simpler than having 2 independent volumes. It works both ways too, if using volume buttons on the speaker then the iPhone volume slider updates instantly. Its similar to how using line-out was better than headphone for connecting speakers via a cable in the old days. I'd suggest trying a speaker with that feature and see if it works better for you, like the Anker Sound Core Sport for example. The Amazon Echo and Sony devices don't have it unfortunately. I'm not sure if it remembers the volume level but think it does.

Update:
This feature is called Absolute Volume Control and was added in the Bluetooth spec AVRCP v1.4 so look for devices that support that.