Hall carded a second round of 67 to share second place, with first-round leader Michelle Wie falling seven shots off the pace after a disappointing 76, 12 shots worse than her course record on Thursday.
McAlpine was scheduled to begin working with Thompson at this week's Ricoh Women's British Open, but her team chose to give him an early try.
"I think you've just got to just play with it and not change too much. I'm really proud of myself how I took advantage of conditions this morning".
It didn't start out so well for Wie though.
Wie, showing no sign of the neck spasms that forced her to withdraw from last month's U.S Women's Open, posted nine birdies during a morning round played in nearly flawless weather on the eastern coast of Scotland.
"I actually hit it the same throughout both nines", she confirmed. "It was definitely a rocky start, but I think that got me a little pissed". "I was angry after that, so I nuked a drive on the next hole and that got me into a good position".
"I really didn't know what I was shooting".
Miyazato, whose slow, simple swing makes her one of the cleanest ball-strikers in women's golf, is making her 14th consecutive appearance at the tournament. She also had a pair of bogeys that left her at even par.
She ended her day with back-to-back-to-back birdies earning her the top spot on the leaderboard. "I'm really happy to be in the team, and it's come around so quickly but I'm just going to focus on next two days, and then I can look forward to the Solheim". It's so pretty out there.
"I wish I knew the reason", she said when asked about her good play this year.
Earlier sudden streaks of scoring promoted England's promising young Hall - just a month younger than the more hyped Charley Hull but a with a lot less time on tour - and the probable favourite going into the weekend, the USA's Thompson.
Mel Reid (-5): "What would it mean to me [to win]?"
Two-back of Kim sits Lexi Thompson, whose secret weapon this week has been her caddie Kevin McAlpine, a former looper at Kingsbarns Golf Links, who has guided Thompson around the course like a seasoned veteran.
Woburn, where Ariya won the Women's British Open past year, is not a traditional links layout, but more like a parkland course with heavily tree-lined fairways. "Something completely different and it ranks right up there with the very best that I have achieved". I've been hitting the ball very well and I'm starting to make some putts.