Do I need to get a new lease in my name for existing tenant

[Calvin Rappard]

I just bought my first rental property in the Chicago area northwest suburbs (Cook county). The current tenants have a lease until end of January, which I will honor. In the closing docs, there was a document that signed over the rights of the current lease to me. Should I still do some sort of addendum to the existing lease to have it in my name, or am I okay to leave until renewal in January? (Existing tenants intend to stay on.)

[Calvin Rappard]

@Jonathan Klemm the property is in Schaumburg, 2 bed 1.1 bath, attached garage. Purchased for $230k with 25% down. Rent is $1,725 with existing tenant, a little under market so scope to increase in future, especially if I do a cosmetic update. Comps in area go up to $2,200 for fully renovated units. 

Breaking even after mortgage, taxes, HOA, so not getting capex etc. on top right now. But as my first one I learnt a lot doing this deal so hoping to use that for future deals.