We consider the self-diffusiophoresis of axisymmetric particles using a continuum description where the interfacial chemical reaction is modeled by first-order kinetics with a prescribed axisymmetric distribution of rate-constant magnitude. We employ the standard macroscale framework where the interaction of solute molecules with the particle boundary is represented by diffusio-osmotic slip. The dimensionless problem governing the solute transport involves two parameters (the particle slenderness epsilon and the Damkohler number Da) as well as two arbitrary functions which describe the axial distributions of the particle shape and rate-constant magnitude. The resulting particle speed is determined throughout the solution of the accompanying problem governing the flow about the force-free particle. Motivated by experimental configurations, we employ slender-body theory to investigate the asymptotic limit epsilon << 1. In doing so, we seek algebraically accurate approximations where the asymptotic error is smaller than a positive power of epsilon. The resulting approximations are thus significantly more useful than those obtained in the conventional manner, where the asymptotic expansion is carried out in inverse powers of In epsilon. The price for that utility is that two linear integral equations need to be solved: one governing the axial solute-sink distribution and the other governing the axial distribution of Stokeslets. When restricting the analysis to spheroidal particles, no need arises to solve for the Stokeslet distribution. The integral equation governing the solute-sink distribution is then solved using a numerical finite-difference scheme. This solution is supplemented by a large-Da asymptotic analysis, wherein a subtle nonuniformity necessitates a careful treatment of the regions near the particle ends. The simple approximations thereby obtained are in excellent agreement with the numerical solution.