Your aircraft came with an inbuilt ACS transponder, specifically one that predates Mode S-ES or does not have that feature enabled, and you bought an additional ADS-B transponder such as a Sky Echo because:
Now you have 2 transponders, how do you configure them? Both On? New On Old Off? Old transmit new receive? Ask this question of other pilots and you may get almost as many answers as there are options. Here is my view, with justification and references.
You should fly with your old ACS transponder enabled, and your new ADS-B transponder both transmitting and receiving. However you must first be sure you have the same aircraft serial number entered into both, or you will look like 2 planes about to collide to air traffic control.
Specifically, open the Sky Echo's web page (address 192.168.4.1), click the Sky Echo Setup link, put your aircraft serial number in the ICAO Address (hex) box. Then enable the 1090ES Transmit Enable checkbox.
If you do not have your aircraft's hex serial number to hand you can use the CAA G-INFO service to look it up from your aircraft registration.
Each of the modes of operation of the transponders adds information to the previous layer. Because they are additive the complete stack should be used. Whilst the new ADS-B transponder duplicates most (but not all) of the ACS information, the ACS transmissions are still necessary for legacy ground stations that do not receive or process ADS-B transmissions.
In increasing order of information they are:
A,C and S are only transmitted in response to your aircraft being 'lit' by the sweep of a radar ground station. You can view them as additions to your aircrafts radar reflection. Your position can be determined by the ground station from the time your ACS transmission arrives back at the radar transmitter and knowledge of the timing of the rotation of its radar beam. Another aircraft receiving your ACS transmission cannot determine your position because it lacks the detailed position and timing information of the radar sweep.
This is why ADS-B is important. Because your ADS-B response includes your GPS co-ordinates, ADS-B receivers in other aircraft are able to receive your ADS-B transmissions and from those alone determine your position. The Air to Air nature of ADS-B transmissions is enhanced by the fact that ADS-B transmissions are made periodically without waiting to be triggered by a radar sweep. Thus having an ADS-B transponder gives you regular updates on the position, velocity and type of all other ADS-B equipped aircraft in range.
Wikipedia' introduction to the transponder modes
Confirmation of the safety of transmitting both ADS-B and ACS by the manufacturer of the Sky Echo 2 is here.
CAA report on why ADS-B transmitters were developed and the technical specification they must meet