An improved ionization injection scheme for laser wakefield acceleration using a tightly focused laser pulse, with intensity near the ionization threshold to trigger the injection in a mismatched plasma channel, has been proposed and examined via 3D particle-in-cell (PIC) simulations. In this scheme, the key to achieving a very low energy spread is shortening the injection distance through the fast diffraction of the tightly focused laser. Furthermore, the oscillation of the laser envelope in the mismatched plasma channel can induce multiple low-energy-spread injections with an even distribution in both space and energy. The envelope oscillation can also significantly enhance the energy gain of the injected beams compared to the standard non-evolving wake scenario due to the rephasing between the electron beam and the laser wake. A theoretical model has been derived to precisely predict the injection distance, the ionization degree of injection atoms/ions, the electron yield as well as the ionized charge for given laser-plasma parameters, and such expressions can be directly utilized for optimizing the quality of the injected beam. Through 3D PIC simulations, we show that an injection distance as short as tens of microns can be achieved, which leads to ultrashort fs, few pC electron bunches with a narrow absolute energy spread around 2 MeV (rms). Simulations also show that the initial absolute energy spread remains nearly constant during the subsequent acceleration due to the very short bunch length, and this indicates that further acceleration of the electron bunches up to the GeV level may lead to an electron beam with an energy spread well below 0.5%. Such low-energy-spread electron beams may have potential applications for future coherent light sources driven by laser-plasma accelerators.