Stellar-mass binary black holes (sBBHs) formed in globular clusters (GCs) are promising sources for multiband gravitational-wave (GW) observations, particularly with low- and middle-frequency detectors. These sBBHs can retain detectable eccentricities when they enter the sensitivity bands of low-frequency GW observatories. We study multiband GW observations of eccentric sBBHs that escape from GC models simulated with the MOCCA code, focusing on how low- and middle-frequency detectors can constrain their eccentricities and other parameters. Using Monte Carlo simulations, we generate 10 realizations of cosmic sBBHs by combining the MOCCA sample with a cosmological model for GC formation and evolution. We then assess their detectability and the precision of parameter estimation. Our results show that LISA, Taiji, the LISA-Taiji (LT) network, and AMIGO could detect 0.8 ± 0.7, 11.6 ± 2.0, 15.4 ± 2.7, and 7.9 ± 1.3 escaping sBBHs, respectively, over 4 yr, while LT-AMIGO could detect 20.6 ± 3.0 multiband sBBHs in the same period. LT and AMIGO can measure initial eccentricities with relative errors of approximately 10−6–2 × 10−4 and 10−3–0.7, respectively. Joint LT-AMIGO observations have a similar ability to estimate eccentricities as LT alone.