So, having read the thread, over the last couple of days, I'm going to leave it at just posting my idea of what I think happened, and then others can knock lumps off it as appropriate.
So when LS was refused admission to the Welly Club, her friends put her in a taxi and sent her home. It seems to be generally assumed that the taxi got her there (I couldn't see anywhere whether police had traced and eliminated the taxi driver)
If she got home and found she was locked out, then that would explain her distressed state as observed in the street. (Was there some mention of her actually having got back into her digs and then left again, still without her phone?)
She then decides her only option is to go back to the Welly Club and find her friends. At the point where she's sitting on the bench trying to sober up and/or waiting for a bus, firstly the bearded good Samaritan stops and then, later, someone who may possibly have been the police suspect. For reasons unknown, she accepts this offer of help, then the perpetrator drives her back towards town but at some point tries it on with her and things get out of hand (could this have happened in the park, accounting for the screams?)
Either way, the perpetrator then disposes of the evidence into the water, somehow, despite the difficulty (I don't know whether there are any locations on the South bank of the Humber where there is deep water right up to the shore?)
One thing I don't understand is that the police must have gone all over the current suspect's car with a fine-toothed comb, yet they don't seem to have found any forensic evidence that LS had ever been in there (unless of course, they are waiting for post mortem checks which can now be carried out.)
In the absence of any further information about the cause of death or the state of her clothing and belongings, which are understandably being withheld by the police, a) so they can use information from them in questioning people and b) to weed out the random nutters who confess to crimes they haven't committed, I think it's not going to be possible to progress this without new witnesses, an arrest, or a confession.