1) Air pollution.
I believe the composition of trapped atmospheric gases in Greenland ice cores have be used to detect minute changes in Earth's atmosphere over decades, centuries and millennia, This has been used to chart the rise and fall of various industries over the ages as they pollute the atmosphere.
So your time calibration package should include an automatic air sampler to bring back an atmospheric sample you can test in your laboratory to see how well it fits in with the record of Atmospheric composition changes in past years, decades, centuries and millennia. If the time machine goes back to a time within the range of scientific studies of atmospheric composition from ice core samples you should know the date within a few years or decades.
2) If you have some sort of automatic robotic optical and/or radio telescopes in your time calibration package, they can go outside and measure the important optical and radio sources during the daytime and the nighttime.
The Moon can always been seen when above the horizon, day or night, except when hidden by clouds. The direction to the Moon, and the phase of the Moon, can be measured very accurately. The phases of the Moon change over a period of 29.6306 days.
The planets Mercury, Venus, Mars, Jupiter, and Saturn would always be visible to the naked eye when above the horizon during nighttime, and sometimes would be among the brightest objects in the sky - Venus, Mars, and Jupiter can sometimes be seen during the day.
If the telescope can tell the difference between a planet and a star, it can identify those planets that are above the horizon. So the directions to the planets, and their phases, will show the positions of the planets relative to the Earth, at the time the observations are made.
For each body in the solar system, there is a synodic period, the period between the times when it's angle relative to Earth is the same. For Mercury the synodic period is 115.88 days, for Venus 583.9 days, for Mars 779.9 days, for Jupiter 398.9 days, and for Saturn 378.1 days.
So if the automatic telescopes are able to find the directions to three or four planets, that will show a relatively rare configuration. A length of time that three or four synodic different periods could fit in as integers and not fractions would be a very long period of time in which each synodic period was repeated many times over. And then if that is combined with the phase of the Moon with its own cycle, that becomes much rarer.
Then there are the apparent positions of the four Galilean moons of Jupiter, which would be easy to detect. They have their own synodic periods in relation to each other. That will make the day that your time machine spends in the past even more unique.
So astronomical observations should make it easy to identify the 24 hour period that your time machine spent in the past or the future, for as long as astronomical orbital simulations are accurate.
3) For a much longer time span, the Sun takes about 200,000,000 years to make a full orbit around the galactic center. The center of the galaxy is the first astronomical radio source ever discovered, so it should be easy for a radio telescope to find the direction to it, if it is above the horizon. Several external galaxies, such as the large and Small Magellanic Clouds, M31 in Andromeda, and M33 in Triangulum, are bright enough to be seen with the naked eye and should be easy for a small telescope to detect. The galaxies Centaurus A and M87 are also famous radio sources that should be easy to detect with a small radio telescope.
So it should be relatively easy to detect the directions to both the galactic center and one or more external galaxies, and thus determine the position of the solar system in it's orbit around the galactic center, and the nearest past or future eras that could be.
4) Send a Mars rover type robot out to collect rock samples (different rock samples in different eras, of course, since you don't want time paradoxes). Most minerals contain trace amounts of radioactive isotopes which decay into other isotopes at a rate that can be calculated. You can test the samples in your lab for the ratios of various isotopes, and if the samples are collected from far enough in the past you should be able to calculate the approximate date.
And there are many similar astronomical and geophysical methods that you can used to detect the date on various long, medium, or short, timescales, and to calibrate various methods against each other to improve accuracy.