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should
past of shall
1—used in auxiliary function to express condition <if he should leave his father, his father would die — Genesis 44:22 (Revised Standard Version)>
2—used in auxiliary function to express obligation, propriety, or expediency <'tis commanded I should do so — Shakespeare><this is as it should be — H. L. Savage><you should brush your teeth after each meal>
3—used in auxiliary function to express futurity from a point of view in the past <realized that she should have to do most of her farm work before sunrise — Ellen Glasgow>
4—used in auxiliary function to express what is probable or expected <with an early start, they should be here by noon>
5—used in auxiliary function to express a request in a polite manner or to soften direct statement <I should suggest that a guide…is the first essential — L. D. Reddick>
[M-W's Col. Dic.]

''Should I have any more questions I will contact you.'' is equivalent in meaning to ''If I have any more questions I will contact you.''

should stands for if. But the definitions of should given above doesn't list any such use, so where does this rule come from?
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Hello,

Oxford Advanced Learner's Dictionary has a longer list.
#6 in that list goes as follows:
OALD 6 (formal) used to refer to a possible event or situation:
If you should change your mind, do let me know.
In case you should need any help, here's my number.
Should anyone call (= if anyone calls), please tell them I'm busy.

(I was wondering if Americans would say any of these. Emotion: thinking)
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TanitI was wondering if Americans would say any of these.
I doubt these are often heard anymore. My dad used to use these now and then, but I seldom hear them from the younger generation.

CJ
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Comments  
I am sure this is said quite often among Americans "If you should change your mind..... please do let me know" . I think it's rather formal ."If" and "should" are serving the same purpose in this mood and either one will serve the purpose in my opinion
 CalifJim's reply was promoted to an answer.
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CalifJimI doubt these are often heard anymore. My dad used to use these now and then, but I seldom hear them from the younger generation.
Thanks. Emotion: smile I sort of knew I'd have received this answer.
If they survive, it will most probably be thanks to us non-native speakers forced to do lots of sentence transformations and write dozens of formal letters for our English exams. Emotion: stick out tongue