If one amendment doesn’t explicitly repeal another one, does it still take precedence over it the way a new law can indirectly override older laws?
But muggles like you and I don't get to decide if a conflict exists, that is a matter for the Court in their role in interpreting the constitution. And the Constitution should be read as a complete document, and not a series of separate and independent clauses. There's a general philosophy in interpreting regulations: more specific rules take precedence over general rules. This is what allows the later amendments to carve out exceptions to the fairly general provisions in the Bill of Rights
Nevertheless, interclause conflict exists. For example, the first amendment restrains powers to pass laws that are implicit in Article I. The first amendment doesn't explicitly repeal any part of Article I, but is an obvious and express effort to limit the powers of Congress.
In other cases, the court has ruled that Congress can't use the powers in one clause to avoid the limitations in another clause. For example, the Commerce clause gives Congress wide powers to regulate trade between the states. But Congress can't use the Commerce clause to create a copyright law that grants copyright in perpetuity as that would conflict with the Copyright clause.
Similarly, the fourteenth amendment covers similar legal ground to the fourth (and perhaps the fifth) The fourth protects you from unreasonable searches, the fourteenth requires due process. Nevertheless, the government can't deny you rights granted under the fourth because they are not explicit in the fourteenth.
Another example: the 24th amendment allows for Congress to prohibit poll taxes. But suppose that Congress felt that the only way to prohibit poll-taxes, it was necessary and proper to also prohibit the promotion of poll-taxes. Could it pass such legislation? Amendment 1 says no. And there is no clear and obvious intent in the 24th that it should overrule the 1st.
However, in a clear case in which a later amendment makes a clear and obvious intent to modify an existing provision, the later amendment takes priority. As a simple example of this, many of the amendments 11-27 include an explicit grant of powers to Congress to legislate. These conflict with Amendment 10 (which says that Congress doesn't have further powers) None of the later amendments repeal 10, but the clear and obvious intent is to grant powers, and so the conflict is generally decided in favour of the later or more specific amendment.
Laws are generally written to minimise conflicting interpretations of how they are to be understood in relation with each other. This is just good legal practise. Nevertheless, conflicts in interpretation can arise. Whilst legislative bodies write and pass legislation, that is laws, it is the role of the judiciary to interpret law - that is the courts. Higher courts over-rule lower courts with the Supreme Court being the highest court of the land (although with a proviso, as we saw in the impeachment of President Trump, it's Congress itself that is a court in this exceptional case).
The constitution of the United States is nothing other than a legal document. Hence it is subject to the above.