So for background you will need to read more about ip addressing and ip networks/subnets. The overall concept is that there is a very large range of addresses and it can be divided down into smaller ranges, but the system is based on binary numbers in 4 groups of 8 ‘digits’ because 8 fit in one byte array. An IP network (aka subnet) is defined by its starting ip address and the subnet mask - because from that you can work out where the end is. Another key point is that it is called a subnet mask - because it ‘masks’ - if you write it all out in binary the mask covers up some parts - these must match - the ones that are left are the ones that can change and still be in the same network. Note at this point that “/” e.g. /24 is actually a subnet mask length not the actual mask - the mask for that would be 255.255.255.0

So all of that said you can then either understand how it works so you can work them out, have an appreciation and learn subnet masks etc, or just know it matters and always check/lookup them up.

With all that out of the way we know (or accept) that rules/patterns must be followed. This is important because it means that your example of 1 192.168.1.1/24 - 192.168.1.254/24 being expanded to 192.168.1.1/23 - 192.168.2.254 /23 is NOT possible. You have the correct idea - but the network must start and end on correct powers of 2 which actually means the /23 network is 192.168.0.1/23 - 192.168.1.254 /23. A simple way to understand this is that 192.168.x.x can have 256 /24 or 128 /23 networks so each /23 must start on an even number not odd.

/24 and /23 are just some common subnet mask lengths. There are some (/8,/16,/24) that derive from different IP network classes - but that is a thing of the past. The public internet uses ipv4 ip addressing - and therefore there is an agreed range for private use (lookup RFC1918) and this includes the 192.168 and 10 ranges - which is why they are so common.

I hope all that is some useful background. now for your specific question and example.

This 10.20.170.10/23 - 10.20.170.239 /23 - will be a range of addresses given out by DHCP. it is often known as a pool/range. Technically the network of the scope is 10.20.170.0 to 10.20.171.255 using 255.255.254.0 mask /23 but only part of that has chosen to be given out by DHCP. This is common practice - perhaps addresses 1 to 9 are used for network equipment. 240 to 245 for servers and then the 10.20.171s all saved for future use.

Using what we now know about where subnets start then /23 does include the 10.20.170 and 171 (because 170 is even and the start of a network boundary).
I think you want to increase the number of addresses available for use in dhcp. Exactly how you do this will depend on the DHCP server (e.g. windows server, linux, a firewall, router etc). Generally most DHCP servers need you to define the whole range so you may need to change the pool/range to this:
from 10.20.170.10 to 10.20.171.239 - but then add an exclusion from 10.20.170.240 to 10.20.171.9 if you don’t want those to be used. although you only need to do this if they are in use. it is done differently on some servers so it really depends.

To end with some of your questions:
How can a segment have more that 256 IPs? Well I think you worked this out, different IP subnets/networks are different sizes.
So my question is in the third octet, can it only go up one number up? - it can go up in other jumps depending on the mask length - e.g. the next is /22 1024 addresses so the third octet jumps 4 to the next network. Different subnet masks do different things - make even different octets change e.g. using a mask length of /16 10.20.x.x is the network and the next would be 10.21.0.0

A real simple overview is that if you start with a big network range e.g. 10.0.0.0 /8 you can divide it in half two get 2 networks which means the mask length increaes by one to /9 then you can half these again, the mask increases to /10 etc and keep going. You get to /30 which has 4 addreses only, then /31 which is 2 and finally a /32 which means one single ip address.

Lets not even mention ipv6 !