This third paper in a series on a parabolic-elliptic chemotaxis model studies bifurcation of positive constant equilibria in one space dimension. We consider $$ \left\{ \begin{array}{ll} u_t=Δu-χ_0\nabla\cdot\left(\frac{u^m}{(1+v)^β}\nabla v\right)+au-bu^{1+α}, & x\inΩ,\\ 0=Δv-μv+νu^γ, & x\inΩ,\\ \frac{\partial u}{\partial n}=\frac{\partial v}{\partial n}=0, & x\in\partialΩ, \end{array} \right. $$ where $Ω$ is a bounded interval, $α,γ,m,μ,ν>0$, $χ_0\in\mathbb R$, and $a,b,β\ge0$. Building on earlier results on global existence, boundedness, persistence, and stabilization, we take $χ_0$ as the bifurcation parameter and examine the effects of $β$ and the domain size on the bifurcation. Center manifold reduction yields a coefficient $β_{n_0}$ at a simple bifurcation mode $n_0$ of a positive constant equilibrium. The pitchfork bifurcation is supercritical if $β_{n_0}>0$ and subcritical if $β_{n_0}<0$. Stronger desensitization raises the sensitivity required for loss of linear stability. Bifurcation may be supercritical for small $β$ but is always subcritical for sufficiently large $β$. Thus, desensitization makes patterns harder to trigger and favors abrupt rather than continuous emergence. The bifurcation mode increases with domain size, allowing more complex spatial patterns. Under suitable conditions, we prove unbounded continuation of local bifurcation branches. We also show that large-sensitivity stationary sequences satisfying a peak-gap hypothesis develop spike-like profiles. Numerical continuation and time-evolution experiments illustrate the bifurcating branches and spike-like solutions and suggest unbounded continuation and increasingly sharp stationary profiles.