Titan has an atmosphere consisting mainly of molecular nitrogen and methane. Solar extreme ultraviolet and X-ray radiation and energetic electrons from Saturn's magnetosphere interact with the upper atmosphere producing an ionosphere. This paper describes improvements to earlier models of Titan's ionosphere. In particular, we consider in more detail ion production from solar ionizing radiation for solar zenith angles beyond the terminator, and a higher spectral resolution soft X-ray flux is adopted in the ion production rate calculations. We demonstrate that significant photoionization takes place well beyond the terminator. K-shell photoionization is also included, and this process adds Auger electrons to the ionospheric photoelectron spectrum, which we model using the two-stream transport code. Our calculated photoelectron spectrum shows a distinct Auger electron peak near an energy of 400 eV.