If so, does it mean we should always use quantificational logic since it combines both syllogistic and proportional logic? Also, Is there any other logic which further builds on quantificational logic?
Syllogistic logic is one of the things that one does with quantificational logic, just as one occasionally works with arithmetic of integers when working in real analysis or group theory (both of which generalize integer arithmetic). We do not take special note of the structure of syllogisms, because they are only a special case of the usual rules of inference; because we have more flexible tools, we do not put as much effort into memorizing what syllogistic formulae are valid or invalid, except as examples of valid or invalid reasoning in general.
In the following, I will let x:A denote that an object x takes a property A.
All A are B: ∀x: (x:A) ⇒ (x:B)
Some A are B: ∃x: (x:A) & (x:B);
Some A are not B: ∃x: (x:A) & ¬(x:B);
No A is B: ∀x: (x:A) ⇒ ¬(x:B).
Syllogistic reasoning can be performed as it was traditionally, albeit with extra instantiation and generalization steps, and often using rules of inference such as modus ponens and modus tollens. Here are two typical examples of syllogistic reasoning, where the first two lines are classic syllogistic premisses, and the final line is the classic syllogistic conclusion.
P1. ∀x: (x:Human) ⇒ (x:Mortal)
Premise — gloss: all humans are mortal
Premise — gloss: Socrates is human
UI. (Socrates:Human) ⇒ (Socrates:Mortal)
universal instantiation — gloss: if Socrates in particular is human, then he is mortal
modus ponens — gloss: therefore, Socrates is mortal
P1. ∀x: (x:Rabbit) ⇒ (x:Furry)
Premise — gloss: all rabbits have fur
P2. ∃x: (x:Rabbit) & (x:Pet)
Premise — gloss: some rabbits are pets
EI. (Flopsy:Rabbit) & (Flopsy:Pet)
existential instantiation — gloss: consider Flopsy, a (generic) pet rabbit
UI. (Flopsy:Rabbit) ⇒ (Flopsy:Furry)
universal instantiation — gloss: if Flopsy in particular is a rabbit, then Flopsy is furry
modus ponens (with a conjunctive simplification) — gloss: therefore, Flopsy is furry
CI. (Flopsy:Pet) & (Flopsy:Furry)
conjunctive introduction (with a conjunctive simplification) — gloss: therefore, Flopsy is a furry pet
EG. ∃x: (x:Pet) & (x:Furry)
existential generalization — therefore, some pets have fur
Syllogistic reasoning is therefore a special case of quantificational logic; it is a framework in which one can confidently reason syllogistically, without being limited only to syllogisms. But one should not feel absolutely restricted to using quantificational logic, or any particular logic; it suffices to use some logic which is both effective and reliable for coming to usable conclusions.