As far as I understand the max. cable length applies to the length between devices (e.g. switches, repeaters etc.). Here I found the following citation:
For 10BASE-T, Clause 184.108.40.206 Medium Attachment Unit (MAU) explicitly states "Provides for operating over 0 m to at least 100 m of twisted pair without the use of a repeater."
I'm not too familiar with repeaters in detail, but according to this a switch should work as well. So the following connection:
switch -> 5m stranded to patch panel -> 90m solid-core to next patch panel -> 5m stranded to switch
will have a length of 100m in total, so it's still within the specs, right? And if the max. lenght only applies to devices this should work:
Overview: switch1 <--100mTotal--> switch2 <--100mTotal--> switch3 Detailed: switch -5m-> patch-90m-> patch-5m -> switch -5m-> patch -90m-> patch-5m -> switch