Yes, but it would be dicey, inconsistent, and wouldn't solve the issue as it would be torn apart as soon as the conservatives gained control back. It would also block every bill going forward in a split legislature.
But, to answer your question and in your scenario, the best way to do it would be through a well crafted law through the commerce clause (inconsistencies in law infringe on women in X state) AND the 14th Amendment's protection of individual freedoms as I don't think one by itself would withstand this court (and I'm not convinced a really well drafted law would still be upheld by this court). But it couldn't be a blanket ban on abortion - that would never pass and would be political suicide for any swing districts, pitting the progressives against the moderates in the Dem Party (progressives would want a complete right to abortion, swing districts would have to be more measured - in the case of rape or incest and in the first two trimesters only, for example) or lose their seats.
Your question should have been, "Smax, what are the ways in which the Constitution can be amended?" because that's basically what it would take to get what you asked:
(1) Both houses propose an amendment with a two-thirds vote, and three-fourths of the state legislatures approve. Twenty-six of the 27 amendments were approved in this manner.
(2) Both houses propose an amendment with a two-thirds vote, and three-fourths of the states approve the amendment via ratifying conventions. Only the 21st Amendment, which repealed Prohibition, was passed in this manner.
(3) Two-thirds of the state legislatures call on Congress to hold a constitutional convention, and three-fourths of the state legislatures approve the amendment.
(4) Two-thirds of the state legislatures call on Congress to hold a constitutional convention, and three-fourths of the states approve the amendment via ratifying conventions.
