It turns out there's a pretty formal affirmative answer using the concept of a flat map (dual to sharp maps).

Theorem: Let $\mathcal K$ be a left proper, combinatorial model category with weak equvialences stable under filtered colimits. Then there is another "flat" model structure on $\mathcal K$ with the same weak equivalences and cofibrations the flat morphisms of the original model structure.

Remark: In a left proper model category, every cofibration is flat. So the flat model structure has more cofibrations than the original one (or else coincides with the original). Moreover, the property of being left proper depends only on being the weak equivalences. So the flat model structure is the one with the given weak equivalences and a maximal number of cofibrations.

Remark: A morphism $\emptyset \to X$ is flat if and only if $X \amalg (-)$ preserves weak equivalences. So under the mild condition that the weak equivalences of $\mathcal K$ are stable under finite coproducts, every object of the flat model structure is cofibrant.

Proof Sketch: By Jeff Smith's theorem, you have to check that (1) the flat maps are stable under cobase change, transfinite composition, and retracts, (2) similarly for acyclic flat maps, and (3) every morphism with the right lifing property with respect to flat maps is a weak equivalence. The first two are pretty straightforward diagram chases (though the transfinite composition part does seem to require the weak equivalences to be stable under filtered colimits). For the last one, by left properness every cofibiration in the original model structure is flat, so every morphism lifting against flat maps is a trivial fibration in the original model structure and hence a weak equivalence.

So this begs the question: what are the flat maps in, say, the Bergner model structure? I think I've identitifed some types, but I'm not sure how to systematically identify them all.

EDIT: As Simon Henry points out in the comments, I've been too cavalier with accessibility issues. But since cofibrant generation of the flat maps is the only condition missing in the almost theorem below, one can take any set of flat maps containing generators for the original cofibrations, and its cofibrant closure gives the class of cofibrations for a cofibrantly-generated model structure. In particular, if weak equivalences are stable under finite coproducts, one can ensure that any desired set of objects is cofibrant in the resulting model structure.

It turns out there's a pretty formal affirmative answer using the concept of a flat map (dual to sharp maps).

Almost Theorem: Let $\mathcal K$ be a left proper, combinatorial model category with weak equvialences stable under filtered colimits. Then there is another "flat" model structure on $\mathcal K$ with the same weak equivalences and cofibrations the flat morphisms of the original model structure.

Remark: In a left proper model category, every cofibration is flat. So the flat model structure has more cofibrations than the original one (or else coincides with the original). Moreover, the property of being left proper depends only on being the weak equivalences. So the flat model structure is the one with the given weak equivalences and a maximal number of cofibrations.

Remark: A morphism $\emptyset \to X$ is flat if and only if $X \amalg (-)$ preserves weak equivalences. So under the mild condition that the weak equivalences of $\mathcal K$ are stable under finite coproducts, every object of the flat model structure is cofibrant.

Almost Proof Sketch: By Jeff Smith's theorem, you have to check that (1) the flat maps are stable under cobase change, transfinite composition, and retracts, (2) similarly for acyclic flat maps, and (3) every morphism with the right lifing property with respect to flat maps is a weak equivalence. The first two are pretty straightforward diagram chases (though the transfinite composition part does seem to require the weak equivalences to be stable under filtered colimits). For the last one, by left properness every cofibiration in the original model structure is flat, so every morphism lifting against flat maps is a trivial fibration in the original model structure and hence a weak equivalence.

So this begs the question: what are the flat maps in, say, the Bergner model structure? I think I've identitifed some types, but I'm not sure how to systematically identify them all.