What I was taught in school was that it's acceptable to omit the comma when joining very short independent clauses. What's "short enough" is a judgment call.
I will happily agree with anyone that (a) it doesn't hurt to put the comma in, and (b) whenever the comma definitely makes things clearer, by all means it should be used.
In your particular case, my judgment is that your second independent clause is short enough that you could get away with omitting the comma. (I would include it, though.)
Ordinarily parenthesis should be marked with commas on both sides. But Fowler recommends that the first comma should be left out after a coordinating conjunction if what follows is a simple adverbial phrase—if I interpret him correctly. (Coordinating conjunctions are and, or, but, and possibly some semi-conjunctive adverbs like so, therefore, however, etc.) The comma should be retained before a subordinate clause or when its omission would cause ambiguity. This probably includes conjunctions that are not at the beginning of a sentence. Fowler's advice seems balanced and practical.
Cleopatra appeared poised and composed. But on the inside, she was seething with rage. — Simple adverbial phrase at the beginning of a sentence.
Her only option was to support Mark Anthony once again. But, if he should perish, it would be the end of her. — Not a simple adverbial phrase but a clause.
She knew her cause was lost and, in a fit of final despair, had a serpent brought in to extinguish her life with its venom. — Not at the beginning of a sentence.
The relevant text from Fowler's The King's English follows. He is arguing for general laxity with adverbial phrases, and would generally allow both commas and no commas around them. He goes on to say that, if commas are used, one must never omit one but write the other; however, he gives the above rule as an exception, in order to avoid an abundance of commas, logical though they might be.
Laxity once introduced, however, has to be carefully kept within bounds. It may be first laid down absolutely that when an adverbial clause is to be stopped, but incompletely stopped, the omitted stop must always be the one at the beginning, and never the one at the end. Transgression of this is quite intolerable; we shall give several instances at the end of the section to impress the fact. But it is also true that even the omission of the beginning comma looks more and more slovenly the further we get from the type of our above cited sentence. The quotations immediately following are arranged from the less to the more slovenly.
His health gave way, and at the age of fifty-six, he died prematurely in harness at Quetta.—Times.
If mankind was in the condition of believing nothing, and without a bias in any particular direction, was merely on the look-out for some legitimate creed, it would not, I conceive, be possible...—Balfour.
The party then, consisted of a man and his wife, of his mother-in-law and his sister.—F. M. Crawford.
These men in their honorary capacity, already have sufficient work to perform.—Guernsey Evening Press.
It will be observed that in the sentence from Mr. Balfour the chief objection to omitting the comma between and and without is that we are taken off on a false scent, it being natural at first to suppose that we are to supply was again; this can only happen when we are in the middle of a sentence, and not at the beginning as in the pattern Cranmer sentence.
Best Answer
The question that you brought up wouldn't address your question because "if" is not one of the coordinating conjunctions, but it is a subordinating conjunction. The words that introduce each are:
Coordinating Conjunctions: and, but, or, yet, for, nor, so (a helpful acronym is FANBOYS)
Subordinating Conjunctions: after, although, as, as if, as long as, as though, because, before, even if, even though, if, if only, in order that, now that, once, rather than, since, so that, than, that, though, till, unless, until, when, whenever, where, whereas, wherever, while
The definition of a subordinating conjunction is:
This article from Purdue on coordinating and subordinating conjunctions explains the usage of the subordinating conjunction as follows:
From this, the correct punctuation of "You can call me if you need me" is:
If you were to move the subordinating conjunction to the beginning, however, you would need the comma as follows: