Any saving throw against any spell you cast.

For a target to have disadvantage against a saving throw, the following conditions must be met:

• the hound is within 5 feet of the target
• The saving throw is against the effect of a spell that you cast

These are the conditions. Let us examine several scenarios. Assume the hound is within 5 feet of the target at all times.

Spell cast anytime in the past, new saving throw against the effect.

Say we cast hold person on the target last turn. The target then ends its turn with our hound next to them. Hold person calls for a wisdom save. Let's check our conditions:

• The hound is within five feet of the target.
• Hold person is a spell that we cast.

Therefore the target has disadvantage on the wisdom save.

Spell cast on our turn.

Say we cast hold person this turn while our hound is within five feet of the target. Let's check our conditions:

• The hound is within five feet of the target.
• Hold person is a spell that we cast.

Therefore the target has disadvantage on the wisdom save.

Spell cast on the target's turn.

Say the target attacks us while our hound is within five feet of the target. We use our reaction to cast hellish rebuke. Let's check our conditions:

• The hound is within five feet of the target.
• Hellish rebuke is a spell that we cast.

Therefore the target would have disadvantage on the dexterity save.

Any saving throw against any spell you cast, no matter when the spell was cast.

For a target to have disadvantage against a saving throw, the following conditions must be met:

• the hound is within 5 feet of the target
• The saving throw is against the effect of a spell that you cast

These are the conditions. Let us examine several scenarios. Assume the hound is within 5 feet of the target at all times.

Spell cast anytime in the past, new saving throw against the effect.

Say we cast hold person on the target last turn. The target then ends its turn with our hound next to them. Hold person calls for a wisdom save. Let's check our conditions:

• The hound is within five feet of the target.
• Hold person is a spell that we cast.

Therefore the target has disadvantage on the wisdom save.

Spell cast on our turn.

Say we cast hold person this turn while our hound is within five feet of the target. Let's check our conditions:

• The hound is within five feet of the target.
• Hold person is a spell that we cast.

Therefore the target has disadvantage on the wisdom save.

Spell cast on the target's turn.

Say the target attacks us while our hound is within five feet of the target. We use our reaction to cast hellish rebuke. Let's check our conditions:

• The hound is within five feet of the target.
• Hellish rebuke is a spell that we cast.

Therefore the target would have disadvantage on the dexterity save.

You cast the spell a month ago.

Suppose we have taken the Keen Mind feat and know exactly what time it is at all times. We cast feeblemind on the target and they fail their save. Exactly 30 days later, we conjure our hound and place the hound within five feet of the target. Feeblemind says:

At the end of every 30 days, the creature can repeat its saving throw against this spell.

Let's check our conditions:

• The hound is within five feet of the target.
• Feeblemind is a spell that we cast.

Therefore the target would have disadvantage against the intelligence save.

Okay, this one may seem a little silly, and a DM may rule that there are some problems with lining up the five minutes of your hound with exactly 30 days after you initially cast feeblemind. But it is an important example because of the language it uses. Notice what it says:

the creature can repeat its saving throw against this spell.

Saving throws against spell effects for spells that were cast in the past are still saving throws against the spell.

Therefore, when Hound of Ill Omen says "the target has disadvantage on saving throws against any spell you cast", this includes saving throws against spells that are made in the future.