Construction of combinatorial model categories with all objects fibrant

Nikolaus has shown (see Cor 2.21) that every combinatorial model category where all trivial cofibrations are monic is Quillen equivalent to its category of algebraically-fibrant objects, in which every object is fibrant.

The category of algebraically-fibrant objects is a category where every object has specified lifts against generating trivial cofibrations, preserved by morphisms. For example, algebraically fibrant Kan complexes have specified lifts for horns.

There are a ton of papers about what you are asking. Another is the thesis of Richard Williamson (arXiv:1304.0867v1). Also, Valery Isaev has a paper that produces a model structure with all objects fibrant, given some cylinder or path object information (https://arxiv.org/pdf/1312.4327.pdf). The thesis of Remy Tuyeres produces a model structure given some even more general category theoretic information. Maybe check the references of those three sources; I'll bet there are many others.