Self-injection locking (SIL) of a diode laser to a high quality-factor microresonator has recently become increasingly important in hybrid integrated photonics, providing access to compact sub-Hz linewidth lasers. It was also shown to facilitate the access to dissipative Kerr solitons - the key to a low-noise coherent frequency comb on a photonic chip. However, the existence and stability ranges of SIL soliton states in experimentally controlled parameters are still not fully understood. Here we study the bifurcation structure of solutions in a model of soliton SIL in the weak-backscattering limit. We show that SIL produces soliton-number-dependent existence ranges of multi-soliton solutions in free-laser detuning and feedback phase parameters. We identify exclusive single-soliton existence regions and demonstrate dynamical access to single solitons in this region by direct numerical simulations using prescribed parameter sweeps.