Two historical precedents, both involving federal judges, make clear that the Senate could also vote to disqualify the president from holding office in the future, with only a simple majority needed.
From Govinfo:
The Constitution provides for removal from office on conviction and also allows the further judgment of disqualification from holding further office. U.S. Const. art. I, Sec. 3, cl. 7. No vote is required on removal following conviction, since removal follows automatically from conviction under this constitutional provision. Deschler Ch 14 Sec. 13.9. However, the further judgment of disqualification from holding future office requires a majority vote. Deschler Ch 14 Sec. 13.10. The Senate has held that a question on removal and disqualification is divisible. 3 Hinds Sec. 2397; 6 Cannon Sec. 512.
https://www.govinfo.gov/content/pkg/GPO-HPRACTICE-112/html/GPO-HPRACTICE-112-28.htm
