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If you have a valid, unexpired multiple entry U.S. visa in your passport, you may enter Mexico for tourist, business or transit purposes for up to 180 days, but no later than expiry of the U.S. visa, without needing a Mexicana visa.
The only thing matters to Mexican authorities is whether there is a physical sticker (visa) within validity period (and not otherwise invalidated) issued by the United States. Whether you have used the visa or not, or if there are supplementary rules on the use of your visa by the U.S. government, does not affect your visa-free eligiblity to Mexico.
Since your H1-B visa is unexpired and there is no mention of valid-from date (even though U.S. immigration rules link the use of this visa to your I-797), you may enter Mexico without a Mexican visa.
Your legal status in the U.S., granted by a border officer on entry, is a distinct concept from a U.S. visa, which only authorizes you to travel to the port of entry to seek admission.
However, Mexico only accepts a valid U.S. visa as a condition of visa waiver; your legal status in the U.S. is irrelevant.
If you have a valid and unexpired visa of the following countries: Canada, the United States, Japan, the United Kingdom of Great Britain and Northern Ireland, and any of the countries of the Schengen Area, you do not need a visa to enter Mexico for tourism, business or transit purposes. All individuals in this category are required to present their valid and unexpired visa along with their passports. Both documents must be valid during your entire stay in Mexico.
If the US visa stamped in your passport is expired but you have an official document or form proving your legal status in USA, you must apply for a Mexican visa at this Consulate. All EAD cards, I-797 A Forms, Advance Parole Forms, I-20 OR DS-2019 are not accepted to enter Mexico.
https://consulmex.sre.gob.mx/sanfrancisco/index.php/servicios-para-extranjeros