We examine which information on the early cosmological history can be extracted from the potential measurement by third-generation gravitational-wave observatories of a stochastic gravitational wave background (SGWB) produced by cosmic strings. We consider a variety of cosmological scenarios breaking the scale-invariant properties of the spectrum, such as early long matter or kination eras, short intermediate matter and inflation periods inside a radiation era, and their specific signatures on the SGWB. This requires to go beyond the usually-assumed scaling regime, to take into account the transient effects during the change of equation of state of the universe. We compute the time evolution of the string network parameters and thus the loop-production efficiency during the transient regime, and derive the corresponding shift in the turning-point frequency. We consider the impact of particle production on the gravitational-wave emission by loops. We estimate the reach of future interferometers LISA, BBO, DECIGO, ET and CE and radio telescope SKA to probe the new physics energy scale at which the universe has experienced changes in its expansion history. We find that a given interferometer may be sensitive to very different energy scales, depending on the nature and duration of the non-standard era, and the value of the string tension. It is fascinating that by exploiting the data from different GW observatories associated with distinct frequency bands, we may be able to reconstruct the full spectrum and therefore extract the values of fundamental physics parameters.