using "should" in the second and third conditional sentences

Question

If you should come across something weird, you would call me.( the second conditional sentence)

Could we use "should" to express a more doubtful idea in the second and third conditional sentences? would you please throw light on such an issue?

Updated: For clarification, I have just put some related examples that illustrate what I am strongly seeking for. You see, therefore, not only as to zero conditional sentences also the other types of conditional sentences you would use should to express more doubtfulness.

Edited: Is there any relation between the sentence R and probability or one of the following?

R. In case you (should)need to me, call me.

1.If I worked harder, I would/should pass the exam. (type 2)

2.If I had worked harder, I would/should have passed the exam. (type 3)

3.if we had lots of money, we would/should travel round the world. (Type 2)

4.If we had had lots of money, we would/should have travelled round the world (type 3)

What is more:

I am wondering what is the difference between the following?

If we had had lots of money, we would/should have travelled round the world

Many thanks

Answer 1

A conditional sentence has two parts: the "if-part", or protasis, and "then-part", or apodosis.

In the apodosis, we use would to create that "conditional" feeling. So the sentence

If we had had lots of money, we should have travelled round the world.

would be illogical: the protasis clearly says that we did not have money during some period in the past, but the second half of the sentence is not an apodosis, because it uses the wrong modal verb. The second half looks like a normal sentence on its own:

We should have traveled around the world! (instead of doing some other things in the past, we should have traveled around the world)

This clause implies that we had the ability to travel around the world. It contradicts the protasis.

The same applies to your conditional 2 example:

If I worked harder, I should pass the exam. [improbable situation in the present or future],[moral obligation]

The first part invites some apodosis with would. The second part just plainly states your obligation to pass the exam. But according to Wikipedia,

Occasionally, with a first person subject, the auxiliary would is replaced by should (similarly to the way will is replaced by shall). (Wikipedia says this about the apodosis of both the second and the third conditional)

So maybe we can use should there after all, since I is a first-person subject. Let a native speaker decide.

The use of should is usually "deontic" (what should be: used to express norms, expectations, speaker's desire) while the use of would is "epistemic" (what may be).

The modal verb should could be sometimes used in the epistemic mood too, so we can come up with sentences such as

If I worked harder, that should be surprising!

But the effect would be comical and ironical, not the usual effect with the conditional sentences. The two halves of the sentence would still look somewhat disjointed.