It's quite hard to see from the photos what exactly is going on with some of it. Is there more than one white cable / set of pairs connected at the moment to the external socket?
Firstly what I would do is neaten up the external box. I would lift the clear cap that has 2,3,5 embossed on it and straighten out all the coloured pairs but still keeping each coloured pair in its own twist. Once you have them all straighted out, cut some of it back so they're all the same length, 6 inches of the twisted pairs in addition to the existing white outer casing should be ample. Then tease out the blue pair from the rest and bend the others over on them self and give them a twist in the middle. Now all the ones not in use are in a neat bunch and you can just work with putting the blue pair back into 2 & 5 connections as before. You'll need to strip back a small bit of the outer casing on the blue pair before placing them back into the 2 & 5 connections but you only want to strip just enough as necessary, not how it is at the moment with the copper showing proud of where the connection actually takes place. Connection 3 and the orange wire is optional, it's the bell wire and isn't often used anymore. Keep the orange pair intact with the rest of the coloured pairs in case someone wants it in future.
One thing that seems a little odd is you say you have 5 or more internal telephone sockets. However, the hallway socket doesn't look as though it has this many extensions so someone seems to have been somewhat creative it seems. It's like someone has given you a mix of a star topology where some extensions feed from the hallway socket and I presume some of your other extensions must be spurred off of socket(s) that are fed from the hallway socket.
When I moved in to my home the previous owner had wired an extension from the outside master as they had put their TV in the opposite corner of the lounge and needed a telephone extension for their Sky box. This corner of the lounge is on the outside wall so it was the best way for them to do it for cosmetic reasons as there was no visible cabling internally. The socket worked fine and I think in many ways it's better than a typical extension that is usually daisy chained off the previous as that adds to the number of joins. I have since removed it because personally I didn't like the cable running around the outside of the house since that side of our property is a shared path and the cable was accessible at ground level to possible malice interference.
The other thing I meant to say is that with the number of extensions you mentioned does that not mean the bedroom next to the landing cupboard already has an extension? If that is the case then surely it would be neater to install an additional extension socket next to that existing bedroom one and then run a shorter length of trunking / cable from there to the cupboard.
Back to the internal extension nearest the front door. I would replace that faceplate with a filtered faceplate like the one you linked to which is a combo master socket and filtered faceplate. You shouldn't see any copper core on the coloured pair connections as they make their connection when punched down into the IDC connections on the back of the faceplate and at the same time any excess from the twisted pairs going into an IDC connection are trimmed off by the punch down tool (also called a Krone tool).
There are two separate lots of connections on the inside of the filtered faceplate, one is the unfiltered (broadband) service and the other is filtered for telephone extensions. If one of them cables in the extension nearest the front door is for the bedroom next to the upstairs landing cupboard then you can use the unused green coloured pair from that cable to connect to the unfiltered extension connections and then use the blue pair to connect to the filtered extension connections. The blue pairs from the rest of the cables can connect directly to the filtered extensions IDC connections. You should be able to do away with having them brass screw down cable connectors being used. If you don't want so many blue pairs connecting to the filtered extension IDC connections then you can get some gel crimp connectors and crimp each of the individual blue pair wires together first before taking the final one on to the IDC connections on the faceplate.
In the bedroom telephone extension socket you could do with a new back pattress box if you want both sockets to be flush to the wall. You can get a dual single gangs pattress box which will have a central divide to provide another set of screw holes so you can attach two single faceplates to it. If the socket is on an outside wall then the pattress box will be metal, otherwise if it's an internal hollow plasterboard wall then it will be a plastic / dry lined type pattress box. Using one of these you can tease out the green pair from the existing cable and poke it through to one side of the new pattress box. The existing blue pair would be fed into the other side of the new pattress box. You can then attach the blue pair to the existing faceplate and attach this to the new pattress box. Buy a new RJ11 faceplate for the other side and connect the green pair to this. Get a decent quality twisted pair cable with a RJ11 connection on one end and whatever connection you need on the other to suit your modem or modem/router and lay this between the new socket and wherever suits best inside the hall cupboard. If one of the walls in the cupboard is an outside wall then I would favour this to attach a wall mounted network rack as you'll be able to use some coach bolts and associated plastic wall plugs, which hold the gear better than the internal partition walls will. If all 3 sides of the cupboard are partition walls then I would buy a small network cabinet that can be ceiling mounted but you will probably have to fix some wooden battening across the ceiling first to span the ceiling joists and so that the wooden battening lines up with the mounting holes of the cabinet.
Or
If your existing pattress box is deep enough you could replace the faceplate on the bedroom extension with a single gang dual euro slot faceplate, this has a big enough hole to accept 2 euro modules, 1 rj11 and 1 telephone extension, like these:
https://www.run-it-direct.co.uk/RJ11module.html