The stock Level 1 EVSE will only charge at the maximum amperage the unit is designed for. There is no way to force it to charge faster.
You should look into the Clipper Creek Level 2 unit. It is about $500 and will charge an empty battery to full in ~4 hours. 240V doubles the speed and 32A is a lot more - the total time is ~5.66X quicker.
Yes I'm planning on getting a L2 charger too, but I'm asking specifically about using a different EVSE that is capable of utilizing a 120 volt / 20 amp outlet and which can charge at 16 amps . This would be useful in circumstances where I do not have 240 volts available. For example at work we have 120V/20A outlets but the stock L1 EVSE can only use 12 amps because most outlets are 15A maximum.