The closer you can test/replicate real working conditions, the better data you'll get if someone is right for the position. Something we started doing that gave us the most relevant data for our efforts was the white boarding challenge where its meant to be highly collaborative. Its not just the interviewee monologuing and asking questions. The interviewers are meant to engage equally as partners to complete the challenge. In a real work situation you're meant to collaborate. So it helps to see how they collaborate, think, take feedback, give or take direction, "yes and" ideas, etc.

In addition we make sure that they are fully prepared for the meeting. We tell them exactly who they will meet with, what will be discussed and what we are hoping to learn about them in that meeting. We include some tips as well. The first onsite meeting is "de-pressurize" and prep them. They meet with other designers, they talk about life, get setup, and give advice. Its meant to put them at ease which of course they are trying to impress us but I've found that it takes the edge off the presentation

Also super important to ensure everyone talking to the interviewee knows everything about them, their work, and what they are suppose to ask and prod for.

Good interview questions should be preset to remove bias and increase quality.

Have a debrief that same day if possible.