In this paper, we study multiple pole solutions for the focusing Hirota equation under the nonzero boundary conditions via inverse scattering method. The direct scattering problem is based on the spectral analysis and exhibits the Jost solutions, scattering matrix as well as their analyticity, symmetries and asymptotic behaviors. Compared with previous studies, we define a more complex discrete spectrum. The inverse scattering problem is explored by solving the corresponding matrix Riemann-Hilbert problems. Particularly, we solve the scattering problem by a suitable uniformization variable on the complex $z$-plane instead of a two-sheeted Riemann surface. Finally, we deduce general formulas of $N$-double pole and $N$-triple pole solutions with mixed discrete spectra and show some prominent characteristics of these solutions graphically. Our results should be helpful to further explore and enrich breather wave phenomena arising in nonlinear and complex systems.