Here is a streamlined description of the optimal strategies that have been found. In both strategies, there is a team that plays a special role -- call them $x$ -- and the other teams are numbered modulo $n-1$.
With $n=2k+2$ teams: We play $2k+1$ rounds of $k+1$ games each. In the $i$-th round, the games are $$(x,i), (i-1,i+1), (i-2, i+2), \ldots, (i-k, i+k).$$ Team $x$ always waits $k+1$ games between playing; every other team either waits $k$ or $k+2$ games.
With $n=2k+1$ teams: We play $2k$ rounds which are alternately "long" ($k+1$ games) and "short" ($k$ games), starting with a long round. In the $i$-th long round, the games are $$(x,i), (i+1,i-1), (i+2, i-2), \ldots, (i+k-1, i-k+1), (i+k, x).$$ In the $i$-th short round, the games are $$(i,i+1), (i-1, i+2), (i-2, i+3), \ldots, (i-k+1, i+k).$$ Every wait is either $k$ or $k-1$ games.
I wish I had time to make some animated graphics of these. I'd put players $1$ through $n-1$ around a circle and $x$ at the center, and draw lines between the pairs as they occur.