This is possibly one of the easiest problems to overcome when taking measurements. This simple solution is to wind the cable being measured around the current clamp a number of times.
The way this works is: For every time the wire passes through the centre of the clamp, the magnetic flux is increased by the value associated with the current flowing in the wire. As the current is the same throughout the wire the flux will be equal to that which is associated with the current multiplied by the number of turns.
The idea is to increase the value being measured by the clamp such that it falls within the usually suggested 10-100% of full scale range. There is one catch (isn't there always!?). The impedance of the clamp, as seen on the wire, will increase by the square of the number of turns wound through the clamp.
Very few clamps operate into a "zero ohms" input. Most operate by generating a voltage across a resistor strapped across the "pickup" secondary winding. In low current clamps (1-10amps) this resistor can start becoming a significant value, even when divided by the ratio of the clamp.
Typical output values are 1V/A with burdens of about 10mW. For a 1 amp clamp this translates into a 1:100 ratio feeding a 100W burden resistor. The primary impedance would therefore be 0.01W. If 10 turns were wrapped around the clamp (to offer a 100mA FSD) it drops the ratio to 10:1 making the impedance on the primary 1W. Although not a huge problem it is at least one that should be borne in mind, especially if using clamps on secondary circuits of primary (also known as 'main') CTs.