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This question presumes that after reading short introductions, the best next step is to read introductory textbooks from each branch of philosophy. Feel free to challenge this presumption.
Then the problem is to select which branches. So which branches of philosophy most:
1. pertain to (English) law, jurisprudence, philosophy of law?
2. most benefit law students and academics?
Please feel free to rank them. (I suspect the relevance of Ethics and Logic, but what else?)
Well, you can probably get the best answer for your purposes simply by googling. There is an entire branch of philosophy called "philosophy of law," which runs from Plato to Grotius and Hobbes to Dworkin and Rawls. Many texts and anthologies are devoted to the subject
It is closely related to philosophy of political science and ethics, as you note. I would not say it is related to "logic" per se, especially not modern logic, but more to argumentation and rhetoric, as developed by the Sophists. The philosophy of religion is also somewhat related, in that most monotheistic religions have foundational "law givers," such as Moses or Mohammed or, to stretch the point, "We the People."
As an aside, I would note that philosophy itself could be described historically as a "branch" of law, or at least as a descendent of the forms of competitive, public argumentation developed earlier in the law courts of the Greek polis and other "political" assemblies of government. Socrates, like Jesus, came to his finale in a law court.
And I would add that in terms of crucial shifts in the modern age, both Kant and Hegel are very important here for developing the modern complexities of self-imposed human "laws" and the discovery of human "freedom." The literature is vast, so any number of overview texts might be a good starting point.
My understanding is that there are 5 major branches of philosophy
epistemology
esthetics
ethics
politics
metaphysics
of which 3 I think would be of most importance
epistemology
ethics
politics
and, although not considered a branch of philosophy, I would include
psychology
as being of great value.
