The best procedure is the one that is right for the particular patient. It must be determined based on testing, things like how strong is the swallow, how long is the esophagus. Then there are factors like is there a hernia? How large and what kind?
As patients, we are foolish to demand specific surgeries just because they are popular. Ask the surgeon what he would do for you.
I had a Toupet (another partial Nissen variety). It was right for me. I asked about the "alternative" types, but my surgeon said they don't do them anymore. They tried them but get better results with Nissen varieties.
You might want to go to a large university hospital, a big place that does thousands of repairs. Not every surgeon will tackle a re-do.