Después del museo, volvimos al palacio para ver la estatua con otra luz.

Because después is normally followed by de:

• después de = after

So:

• después de el museo would be the full form,
• but de + el contracts to del.

That is why you get:

• después del museo

This contraction is mandatory in standard Spanish.

In natural Spanish, después del museo often means after the museum visit or after our time at the museum.

Spanish often uses a place noun this way when the activity is obvious from context. So this sentence does not usually mean that the museum itself ended; it means the people were done being at the museum.

If you wanted to be more explicit, you could say:

• Después de visitar el museo...
• Después de salir del museo...

But después del museo is completely natural.

Volvimos is the preterite (simple past), first person plural:

• volver = to return / to go back
• volvimos = we returned / we went back

Here it refers to a completed action in the past: they went back to the palace.

A quick comparison:

• volvemos = we return / we are returning
• volvimos = we returned

Because Spanish usually drops subject pronouns when the verb ending already makes the subject clear.

• volvimos already tells you the subject is we
• so nosotros is unnecessary unless you want emphasis or contrast

For example:

• Volvimos al palacio = normal, neutral
• Nosotros volvimos al palacio = more emphatic, maybe contrasting with someone else

This is very common in Spanish.

Because a + el contracts to al.

So:

• volver a
  • a place
• a el palacio becomes al palacio

This is another mandatory contraction in standard Spanish.

The two main contractions you must remember are:

• de + el = del
• a + el = al

That is why this sentence has both:

• después del museo
• volvimos al palacio

Because when volver means to return/go back to a place, it is followed by a:

• volver a casa
• volver al hotel
• volver al palacio

So volvimos al palacio means we went back to the palace.

Be careful: volver a + infinitive can also mean to do something again:

• Volvimos a ver la estatua = We saw the statue again

But in your sentence, the structure is different:

• Volvimos al palacio para ver la estatua
• We returned to the palace in order to see the statue

Para + infinitive is a very common way to express purpose:

• para ver = in order to see / to see

So the sentence structure is:

• volvimos al palacio = we returned to the palace
• para ver la estatua = in order to see the statue

This tells you why they returned.

A very common pattern in Spanish is:

• ir / venir / volver / salir + place + para + infinitive

Examples:

• Fuimos al mercado para comprar pan.
• Volvieron a casa para descansar.

Because para is the normal preposition for purpose.

Here, the meaning is we returned to the palace in order to see the statue, so para ver is correct.

Very roughly:

• para = for, in order to, with the goal of
• por = because of, through, by, around, in exchange for, etc.

So in this sentence:

• para ver = to see / in order to see

Using por ver here would sound wrong.

Literally, it means with another light. In context, it usually suggests one of these ideas:

• under different lighting
• in a different light
• with the light changed, perhaps later in the day

It can be partly literal and partly stylistic. For example, they may have gone back when the sunlight was different so the statue looked different.

So con otra luz is a very natural Spanish way to express the idea that the statue would be seen differently because of the lighting.

Because otro/otra normally does not take un/una before it.

So Spanish says:

• otra luz
• otro día
• otra vez

not:

• una otra luz
• un otro día

This is an important difference from English, where another already includes the idea of an + other. Spanish otro/otra works similarly.

Many adjectives do come after the noun, but some common determiners and adjectives usually come before it. Otro/otra is one of them.

So:

• otra luz
• otro libro
• otra estatua

This is normal word order.

In other words, otra behaves more like a word such as this, that, or another in English than like a typical descriptive adjective.

Spanish usually needs an article with singular countable nouns in sentences like this.

So:

• ver la estatua = to see the statue

Using ver estatua would be ungrammatical here.

Also, la suggests that the statue is a specific one already known from the context, probably a particular statue in or near the palace.

Yes. Luz is a feminine noun:

• la luz
• otra luz

In Spanish, grammatical gender is not determined only by the final letter. While many feminine nouns end in -a, plenty of feminine nouns do not, and luz is one of them.

That is why the adjective agrees in the feminine form:

• otra, not otro