right off the bat

The Partnerから最後の引用です。
Sandy leaned a bit closer. "We're prepared to offer twenty-five thousand dollars. Right now. The check's in my pocket."
She was leaning down too, getting lower and closer to his face, when the money hit and stopped her cold. Her eyes watered and her bottom lip quivered. "Oh my God," she said.
Sandy glanced around. "That's right, twenty-five thousand bucks."
She ripped a paper napkin from the holder and in doing so knocked over the salt shaker. She dabbed her eyes, then blew her nose. Sandy was still glancing around, hoping to avoid a spectacle.
"All mine?" she managed to say. Her voice was hoarse and low, her breathing rapid.
"All yours, yes."
She wiped her eyes again, then said, "I need a Coke."
SHE DRANK a 44-ounce Big Gulp without a word. Sandy sipped his bad coffee and watched the foot traffic come and go. He was in no hurry.
"The way I figure it," she finally said, clear-eyed now, "is that if you walk in here and offer twenty-five thousand right off the bat, then you're probably willing to pay more."
"I'm in no position to negotiate."
上に出てきた慣用句らしき表現の "right off the bat" を調べます。
・Oxford English Dictionary: At the very beginning.: Well, let me say right off the bat that I am pretty impressed.
・McGraw-Hill Dictionary of American Idioms and Phrasal Verbs: Fig. immediately; first thing. When he was learning to ride a bicycle, he fell on his head right off the bat.: The new manager demanded new office furniture right off the bat.
"right off the bat" の意味は分かりましたが、語源が気になるので更に調べると、"Know Your Phrase" に次の説明がありました。
In baseball, this phrase references the ball coming off the bat after a successful strike, which is then immediately followed by the batter making a quick decision to run towards first base. The immediate response taken by batters after a successful swing would be my guess on how this saying got its figurative meaning of doing things quickly, or without delay.