Yes, and this is one of the main results in a paper I hope to put on arxiv very soon. I wrote about this result in a previous mathoverflow answer [here][1]. You are right that the way to do it is to focus on maps between cofibrant objects. This result was also known to Clark Barwick, and probably to Cisinski, as I discuss in my other answer. That's one of the reasons it took me so long to write it up. I didn't know why anyone would care. But, in the paper I'm finishing, there are tons of applications of this result. If you want to correspond more, I would be happy to.

Edit in response to request from the OP:

The conditions of the theorem state that $M$ is locally presentable, and 

1. $W$ is $\kappa$-accessible for some $\kappa$, 
2. $W$ is closed under retracts, 
3. morphisms in inj$(I)$ are weak equivalences, 
4. within cof$(I)\cap W$, morphisms with cofibrant domain are closed under pushouts of diagrams of cofibrant objects, and are closed under transfinite compositions,
5. The maps of $I$ have cofibrant domain and the initial object of $M$ is cofibrant.

Then you have a combinatorial semi-model structure with generating cofibrations $I$, generating trivial cofibrations $J$ constructed as in Barwick or Beke's papers, cofibrations cof$(I)$, and fibrations rlp$(J)$.

I should be done with this paper by the end of 2019. Right now I'm just adding lots of examples.

  [1]: https://mathoverflow.net/questions/300879/left-bousfield-localization-without-properness-what-is-known