The question presented in the title is a bit complex, because neither the Order of the Temple of Solomon nor the European authorities were perfectly united and thus persecutions of the Order in the early 14th century did not affect Templars in all countries equally. But let's begin with saying that the often cited date of 13 October 1307 (commonly mentioned in the context of discussing the superstitions related to Friday the 13th) is not a date of the destruction or dissolution of the Order. On that day, the French king Philip IV issued an order to arrest all Templars in France and only in France (obviously having no jurisdiction outside his land, although majority of Templar possession was located in the French area). This was, of course, a major event that precipitated the ultimate downfall of the Order, but even though the French king could have make a decision concerning the operation of various organizations in his country, he was unable to completely eliminate the Order, as it was the sole prerogative of the papacy. The Templars were officially disbanded only by the papal bull Vox in excelso that has been presented to a Council of Vienne on 22nd March 1312 and officially adopted on 4th April 1312. This document was followed in the same year by the bulls Ad providam (2nd May) and Nuper in concilio (16th May) that mandated the transfer of all holdings and possession of the Templars to the Joannites.

The date you mentioned (i.e. 'late October or November') most likely refers to the papal bull Pastoralis praeminentie that has been issued on 22nd November 1307, slightly over a month after the arrests in France and urged all secular rulers to follow the example of Philip IV and arrest the Templars in their lands, effectively legitimizing the actions of the French king and enacting the process of the dissolution of the Order that was formalized during the aforementioned Council of Vienne.

It should be noted, however, that Templars in the Iberian Peninsula developed strong ties to secular rulers and eventually pledged their loyalty to them. In Portugal, this allowed them to reform into a nominally different order (Military Order of Christ) even though its core was formed by former Templars. The rulers of Castille and Portgal followed with arrest of the Templars and the confiscation of their assets, although the knights themselves were eventually acquitted of any crimes in the Council of Salamanca held in 1310, while rulers of Aragon held similar council in Tarragona in 1312, with more or less the same results. The possessions of the Templar in Aragon were distributed between other already existing orders, e.g. Brothers of Caceres, Knights of Santiago, Order of Montesa, Order of Calatrava or the Order of Alcantara, with some of the possessions being transferred to Joannites. But even though the Templars were not persecuted as they were in France and many continued their service in other knightly orders, the Order of the Temple of Solomon as such ceased to exist nonetheless.

So, to sum it up, the dissolution of the Templar Order was a process that took several years. Initiated by Philip IV on 13th October 1307 as a secular action directed against Templars in France, was then followed by papal approval on 22nd November and resulted in seizure of assets and arrests of Templars across Europe, but the Order as such has only been officially disbanded four and a half years later, on 4th April 1312.