Will replacing 10/100 switches with GB switches make our network problems better or worse?

We have about 4 locations connected to our branch via Router to Router VPN all users in outside locations use RDP to connect to our Terminal server for business applications and use Outlook on their desktop. We have our servers connected to a gigabit switch which is daisy chained off another gigabit switch that is connected to the gigabit port on our router. We have some users plugged into the gigabit switch closest to the router. then we have 2 more 10/100 switches daisy chained one off another from the gigabit router a 24 and 16 port switch. The sales has a 24 port switch near them and then engineering has a 24 port switch near them all 10/100. The warehouse has a 16 port switch near them all daisy chained off the 24 10/100 switch closest to the router. Sales, Engineering and the Warehouse have very slow ping times averaging 230milli. Some people can't even connect to the network because they can't reach the dhcp server for an address. I'll be improving the topology of it soon but was wondering would replacing all 10/100 switches not only fix our problem but improve speed or would it make problems worse because they can all connect to the servers at a faster speed therefor slowing things down even more? I'm guessing the main problem is the 24 port old 3com 10/100 switch that we have but I'm to any other idea's and possibilities of the high ping time and connectivity issue.