Did Tom Riddle have The Trace on him?

Perhaps because the time period in which Tom Riddle performed his first acts of crime were concurrent with the time period in which Grindelwald was still gaining power, and from what we have learned in Fantastic Beasts: The Crimes of Grindelwald, there are people like Credence Barebone (Grindelwald's weapon) and Grimmson (a bounty hunter who takes orders from both the British Ministry of Magic and Grindelwald. So, quite literally, an INNER) who posed a threat, a secret threat in the case of Grimmson, against Britain. Who knows, maybe Grindelwald told Grimmson to Imperius other wizards in Britain during Grindelwald's reign? We also knew that even the members of the British Ministry of Magic were afraid of Grindelwald. Grindelwald easily dealt with more than a couple dozen Aurors with ease, and none of his followers were killed by his enemies, save for the girl who got shot with a Killing Curse. Also, none of his followers fought. As such, magical Britain might have been in a similar state to what it was in the First and Second Wizarding War as even the Aurors had a hard time. Also, that similar ministries of magic had fallen already against Grindelwald.

Here's an excerpt from Half-Blood Prince:

— — ISSUED ON BEHALF OF — — The Ministry of Magic PROTECTING YOUR HOME AND FAMILY AGAINST DARK FORCES The Wizarding community is currently under threat from an organization calling itself the Death Eaters. Observing the following simple security guidelines will help protect you, your family, and your home from attack.

1. You are advised not to leave the house alone.
2. Particular care should be taken during the hours of darkness. Wherever possible, arrange to complete journeys before night has fallen.
3. Review the security arrangements around your house, making sure that all family members are aware of emergency measures such as Shield and Disillusionment Charms, and, in the case of underage family members, Side-Along-Apparition.
4. Agree on security questions with close friends and family so as to detect Death Eaters masquerading as others by use of the Polyjuice Potion (see page 2).
5. Should you feel that a family member, colleague, friend, or neighbor is acting in a strange manner, contact the Magical Law Enforcement Squad at once. They may have been put under the Imperius Curse (see page 4).
6. Should the Dark Mark appear over any dwelling place or other building, DO NOT ENTER, but contact the Auror office immediately.
7. Unconfirmed sightings suggest that the Death Eaters may now be using Inferi (see page 10). Any sighting of an Inferius, or encounter with same, should be reported to the Ministry IMMEDIATELY.

For these reasons, the Trace may have simply been suppressed or removed during the Global Wizarding War against Grindelwald. Therefore, Tom Riddle's magic may not have been sensed by the Ministry as they had more trying events in their hands, and by the looks of it, in as early as 1927, the multinational Aurors were already losing.

Although canonical support is sketchy, what we know already from the Trace leaves the following explanations (and perhaps others):

The Ministry does not monitor underage magic use at all when school is in session.

If Riddle did his dirty work at that time, he escapes detection. So he managed to sneak away from the school for a few hours during the night, or perhaps on a weekend. I don't see what he did as taking more than a few hours to accomplish. We already know (from Prisoner of Azkaban) that students can leave the school grounds without being detected.

Riddle found a way to suppress the Trace.

The nearest we come to canonical support for this is the established fact that Riddle was an extraordinarily gifted wizard. If any student found a way to get out from under the Trace, it would be he. Since the Trace is a restriction on his activities, he would certainly be motivated to negate its effects.

Harry asks Dumbledore why the Ministry couldn't detect Tom Riddle's magic:

"But how come the Ministry didn't realize that Voldemort had done all that to Morfin?" Harry asked angrily. "He was underage at the time, wasn't he? I thought they could detect underage magic!"

"You are quite right — they can detect magic, but not the perpetrator: You will remember that you were blamed by the Ministry for the Hover Charm that was, in fact, cast by —"

"Dobby," growled Harry; this injustice still rankled. "So if you're underage and you do magic inside an adult witch or wizard's house, the Ministry won't know?"

"They will certainly be unable to tell who performed the magic," said Dumbledore, smiling slightly at the look of great indignation on Harry's face. "They rely on witch and wizard parents to enforce their offspring's obedience while within their walls."

As for why the Ministry didn't investigate, well they actually did investigate, and the investigation led them to Morfin. As Dumbledore explained:

"The Ministry, on the other hand, knew at once that this was a wizard's murder. they also knew that a convicted Muggle-hater lived across the valley from the Riddle house, a Muggle-hater who had already been imprisoned once for attacking one of the murdered people.

"So the Ministry called upon Morfin. They did not need to question him, to use Veritaserum or Legilimency. He admitted to the murder on the spot, giving details only the murderer could know. He was proud, he said, to have killed the Muggles, had been awaiting his chance all these years. He handed over his wand which was proved at once to have been used to kill the Riddles. And he permitted himself to be led off to Azkaban without a fight.

As discussed previously, The Trace was place-localized, NOT person-localized (e.g. Harry was blamed for Dobby's magic).

In your example, Riddle blamed it on Morphin who was obviously nearby, and therefore wasn't under suspicion.