We studied low-energy plasmons in ultrathin films of silver in the thickness regimes where the surface states as well as quantum-well states must play significant roles. Realistic band structure was adopted for assessing the quantum-mechanical effect on the low-energy charge dynamics. In addition to the expected quasi-two-dimensional plasmon mode, we find the modes that resemble an acoustic surface plasmon on semi-infinite metal surfaces and an additional plasmon mode related to the interband transitions between the two slab-split surface states. It is found that the dispersion of the latter mode is almost identical to the acoustic surface plasmon except the energy offset at small momenta values. The peaks in the surface response function related to interband transitions between the surfacelike states and bulklike states are identified as well. The present work elucidates the role of quantized electronic states and surface states on the plasmonic excitations in the ultimately thin films potentially used in the future nano-optics devices.