The Earth's matter effect is going to play a crucial role in measuring the unknown three-flavor neutrino oscillation parameters at high confidence level in future high-precision long-baseline experiments. We observe that owing to the new degeneracies among the most uncertain oscillation parameters $\left(\delta_{\mathrm{CP}},~\theta_{23}\right)$ and the average Earth's matter density $\left(\rho_{\mathrm{avg}}\right)$ for the 1300 km baseline, the sensitivity of the upcoming Deep Underground Neutrino Experiment (DUNE) to establish Earth's matter effect reaches only about 2$\sigma$ C.L. for all possible choices of oscillation parameters. We notice that the current uncertainty in $\delta_{\mathrm{CP}}$ degrades the measurement of $\rho_{\mathrm{avg}}$ more as compared to $\theta_{23}$. To lift these degeneracies, we explore the possible complementarity between DUNE and Tokai to Hyper-Kamiokande (T2HK/JD) facility with a second detector in Korea, popularly known as T2HKK or JD+KD setup. While DUNE uses wide-band beam with on-axis detector, T2HKK setup plans to use narrow-band beam with two off-axis detectors: one in Japan and other in Korea. We exhibit how the high-precision measurement of $\delta_{\mathrm{CP}}$ in JD+KD setup and the information on $\rho_{\mathrm{avg}}$ coming from DUNE can reduce the impact of these degeneracies in both $(\rho_{\mathrm{avg}}-\delta_{\mathrm{CP}})$ and $(\rho_{\mathrm{avg}}-\theta_{23})$ planes. We show that the combined data from DUNE and JD+KD setups can establish Earth’s matter effect at more than 6$\sigma$ C.L. irrespective of both the choices of mass hierarchy: normal (NH) and inverted (IH), $\delta_{\mathrm{CP}}$, and $\theta_{23}$. With the help of this combined data set, we can measure the average matter density $\left(\rho_{\mathrm{avg}}\right)$ with a relative 1$\sigma$ precision of around 11.2\% (9.4\%) assuming true NH (IH) and $\delta_{\mathrm{CP}} = -90^{\circ}/90^{\circ}$.