Weakness in Pohlig-Hellman algorithm

Let's try to solve a discrete logarithm:

$$\beta \equiv \alpha ^{x} \bmod \,\, p$$

using the Pohlig-Hellman algorithm. Let's suppose that $$p-1=tq$$, where $$q$$ is a large prime number. This means that the resolution is probable infeasible, however, if $$t$$ is factorizable with small prime numbers the algorithm can determine the discrete logs $$\bmod t$$ in an efficient way. But at this point, what i do with the discrete logs $$\bmod t$$?

I am also having trouble understanding this passage:

Note that even if $$p - 1 = tq$$ has a large prime factor $$q$$, the algorithm can determine discrete logs mod $$t$$ if $$t$$ is composed of small prime factors. For this reason, often $$\beta$$ is chosen to be a power of $$\alpha^t$$. Then the discrete log is automatically $$0$$ mod $$t$$, so the discrete log hides only mod $$q$$ information, which the algorithm cannot find. If the discrete log $$x$$ represents a secret (or better, $$t$$ times a secret), this means that an attacker does not obtain partial information by determining $$x \bmod t$$, since there is no information hidden this way. This idea is used in the Digital Signature Algorithm, which we discuss in Chapter 9. The point is that even though $$p - 1 = tq$$ may be large, the discrete log security of $$(\mathbb Z/p\mathbb Z)^\times$$ against Pohlig–Hellman depends on the size of $$q$$, not on the (possibly much larger) size of $$p$$ or $$tq$$. If $$q$$ is the largest prime factor, then the cost of computing discrete logs modulo $$p$$ is essentially at most the cost of computing order-$$q$$ discrete logs in Pohlig–Hellman. This is why, e.g., Schnorr groups are chosen to have an order with a large prime factor.