The following is based on the Amstrad PPC 640S Service Manual. I have never had an Amstrad so this is based on that service manual.
Using the features of the Amstrad and standard features of telphone system circuits it may be possible for you to achieve your preferred configuration.
Starting point is the Amstrad is designed to sit between the incoming telephone line and your telephone handset. As other answers have correctly assumed there is a relay (R901 on Modem Chassis Schematic Diagram 2) that disconnects your telephone handset from the telephone line when the Amstrad Modem is active. This normally prevents the handset socket being used to connect the Amstrad Modem via the handset socket to a telephone line.
Wiring of the telephone system has to be adaptable using different wiring configurations. The Amstrad is designed not to disrupt existing telephone wiring. Using this feature it may be possible to achieve your desired configuration.
Amstrad was a UK designed consumer device, the catch: this might not be legal wiring in your country. Laws and regulations vary between countries and I do not know your circumstances. Your responsibility to check any Legal/Regulatory requirements before modifying any telephone cabling, leads, etc.
Historically telephone systems would use two pair cable for domestic installations. This would permit connection of two separate telephone lines and the telephone sockets are provided to accomodate multiple lines. A standard handset connected to the Amstrad socket will use pins 2 and 5 for a connection to the telephone line. To accommodate other wiring configuration pins 3 and 4 are shown directly wired to the Amstrads telephone line cable. If you were to connect a telephone line to pins 3 and 4 of the handset jack the incoming telephone line should then appear on the connector of the Amstrad telephone lead. Just this step is likely to have broken any number of Laws and/or Regulations in most countries. You may also have an electrical safety issue of the telephone line voltage on the Amstrad telephone lead.
The incoming telephone line now has to be looped back to the Amstrad Modem. This means you need the Amstrad telephone lead plugged into a standalone socket that is appropriately wired. Installing a suitable socket in a small plastic box is one solution.
Best guess from the poor quality Amstrad schematic image I'm using is the telephone cable connection are part numbers:
Amstrad: CD951
Manufacture: AMS45125
With the following information on the Schematic:
P1 GRE (i.e. Green) - Amstrad Telephone Socket Pin 4
P2 YEL (i.e. Yellow) - Telephone line to Amstrad Modem
P3 BLA (i.e. Black) - Telephone line to Amstrad Modem
P4 RED (i.e. Red) - Amstrad Telephone Socket Pin 3
Best guess:
GRE = Green
YEL = Yellow
BLA = Black
RED = Red
Obviously this colour code does not match the colour of the wires in your photograph, nor the pin numbers of a current standard telephone service. The Schematic's Electrical Parts List, Miscellaneous includes:
Circuit Reference:
Description: Modem Extension Cable
Part Number: 177325
There are no details for the wiring of this Modem extension cable. I can not tell from your photograph if the Amstrad telephone extension lead is 2 or 4 wire. If it is 4 wire then the best guess is the Amstrad Telephone lead is wired in exactly the same configuration as the Amstrad Handset socket. If it is only two wire then you are out of luck. The standalone/isolated Telephone style socket (e.g. RJ45) would be wired pins:
P3 to P2
P4 to P5
Use a multimeter to confirm this wiring.