Transformer Flux Linkage

Yes, bifilar winding is very effective indeed in improving coupling and reducing leakage inductance. But it leads to high interwinding capacitance and poor insulation-voltage capability. For off-line transformers there will be safety issues and it's generally not allowed.

Second best in reducing leakage inductance is to interleave the primary and secondary windings, either side-by-side, or one above the other. E.g., P - S - P. If side-by-side, you can have plastic insulating spacers, if on-top you can separate windings with layers of tape. More than one set of interleaving produces better results, but is harder to make.

There are simple formulas to predict exactly what your leakage inductance will be. BTW, the leakage inductance is the same whether the core is present or not. The following is from Snelling, Soft Ferrites, 2nd-ed, pg 337.

There's an 1/M² term in the leakage inductance equation, and you can see the configurations with M = 2, 4 and 6 providing reductions from 4 to 36 times. We won't go into all the problems that poor coupling and excessive leakage inductance can cause, or exacerbate, but it's often well worth any effort spent on improvement. I see the figure is hard to read after CR4's compression, so I've blown up the definitions of terms.