If the Massachussetts convention votes down the 1787 Constitution on February 6, 1788, then they will probably still request that the Rights Amendments are added, as they did when they approved the Constitution IOTL. The difference here is that another ratification convention would be contingent on the addition of those amendments.
Rhode Island will still reject the 1787 Constitution on March 24, 1788, as they did IOTL, only this time they will be adding momentum to the movement to reject the Constitution. I don't think it would be enough to sway Maryland's ratification vote, however, so on April 28 that state would become the sixth to ratify, bringing confidence back to the constitution supporters.
South Carolina will most likely still ratify on May 23, becoming the seventh. Ratifying conventions for New Hampshire, New York, and Virginia will begin at roughly the same time in early June. This will be the critical period that will determine the future of this timeline... will the rejection of the Constitution by Massachussetts influence one of these states to vote it down, where IOTL they ratified the document? Let's see:
IOTL June 21, New Hampshire ratified the Constitution by a 57-47 vote. The state is small in population, so its representatives could more easily swayed by divergences than a larger state. Especially if those divergences were nearby... being in New England along with Massachussetts, I'm guessing this vote could easily go the other way. IOTL, New Hampshire requested along with its ratification the addition of the Rights Amendments. ITTL, I could easily see them following Massachussetts' lead and requesting the Amendments before ratification before they vote on it again.
IOTL June 25, Virginia ratified the Constitution by a 89-79 vote and requested Rights Amendments. ITTL, with Massachussetts rejecting ratification, and especially if there is the added influence of New Hampshire also rejecting it, the vote could easily go the other way. Rights Amendments would still likely be requested.
IOTL July 26, New York ratified the Constitution by a 30-27 vote and requested Rights Amendments. ITTL, with Massachussetts rejecting ratification, and especially if there is the added influence of its rejection by New Hampshire and Virginia, the vote could easily go the other way, with the request of Rights Amendments still made.
With North Carolina refusing to hold a ratifying convention until the addition of Rights Amendments as IOTL, the supporters of the 1787 Constitution would be forces to go through the Confederation to hold another convention in which the constitution could be debated once more, and a new document produced.
IOTL, anti-Federalists by even early 1788 were already arguing for a second convention. The movement to do so reached its climax in September, when Pennsylvania delegates met in Harrisburg to propose amendments and expedite the organization of a second convention. The anti-Federalists had a network that extended to all of the states that were skeptical of the Constitution: New York, the Carolinas, Massachussetts, and others. ITTL, the Harrisburg convention would likely be organized earlier, say in August, shortly after it becomes clear the Constitution will not be ratified. It is possible that this becomes a national affair, rather than a state one. The Harrisburg convention could attract delegates from many states, and they will likely act in order to propose a Second Constitutional Convention to meet somewhere the following year.
Many feared that the anti-Federalists and other populists would use violence to intimidate the Federalists into adopting a constitution they favored. The Carlisle Riots would still have happened here, so those fears would have still been present. With such pressures, all groups would probably want to convene as soon as possible, say March or April 1789, and would probably still choose Philadelphia to avoid political battles over the location. The delegates will be very different, and the situation will be very different as well, with factions much more clearly delineated than before.
The tensions may prove too much for the Second Convention... they'll likely start by adding Rights Amendments, as requested by several states, but there will be delegates that will want to scrap the entire document and start anew, or make major modifications. Worst case scenario, it could fall apart, with delegates instead proposing a Constitution that will fix only the basic problems of the Articles of Confederation that they know could be ratified by all the states. Best case scenario, many Rights Amendments are accepted, some parts of the 1787 Constitution are tweaked, and ratification proceeds afterward, as the states that rejected ratification now have had their demands met.