Splicing into existing light fixtures??

Hello all. First post, but long time lurker. I am in need of some electrical advice. I have a basement living room with 2 existing light fixtures. I am wanting to replace them with recessed fixtures and add 4 more. More specifically, I would like to splice 2 new fixtures to each existing fixture like I drew in the following diagram. Also, the switch is 3-way if that makes a difference.

Is it ok to just splice into these locations, or do they all need to be in line?? Also, the switch is 3-way if that makes a difference.

Splicing into existing light fixtures??