No, you can't have a spur of a spur since the load on 2 x d/sockets could be 13amp x 4 which is OTT, a single TE 2.5mm cable can handle about 27 amps.
If you really can't take a new spur from another (none spurred) socket then you could put the spur cable on a 13 amp fused spur, then feed in to the existing spur socket and then on to a 2nd spur. That would limit the load over all the spurred sockets to 13 amp.
Or you could take a spur from another socket, that has not been spurred from or extend the circuit.
But your solution is as posted above, 13A FCU from original socket-outlet then you can run a number of socket outlets from there on.
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