Short answer: Absolutely, Yes!
This question was posed by one of my students who noted a local airport, Austin Executive (EDC) now has a working control tower, but it isn’t currently noted on the VFR sectional. He correctly noted that FAR 91.126(d) states you have to establish 2 way radio contact with every control tower before entering a 4nm radius up to 2500AGL, but was concerned that you would miss this if you looked at just the sectional.
This is covered by FAR 91.103 which requires you to become familiar with all available information concerning that flight. The Chart Supplement (formally the Airport/Facility Directory – A/FD) information is large part of that information as it includes airport information like runway lengths, fuel availability, traffic pattern information, lighting information, airport attendance information, taxi diagrams, airport layout information, and various airport specific remarks (noise abatement, wildlife, who to call for instrument clearance, etc.). This is all important information for pilot heading to an unfamiliar airport. While checking this information for every airport on the planned route might be excessive, I would certainly (and I am required to) check every airport I was going to land at (including alternates).
In addition, the Chart Supplement is updated every 56 days. VFR sectional charts are updated every 6 months. Lots can change at airport in 6 months (the taxi diagram for GTU still isn’t right – for instance). The last VFR chart that included EDC was published in April 2018 and included information that was received in March 2018. The next VFR chart that includes EDC will be available November 8th – it should include the correct tower frequency. All of this information is included on the legend page of the sectional chart. There was a NOTAM on the tower until the chart supplement was updated to include the tower information.
Fortunately, the good news is that Foreflight (and I am sure the other iPad flight planning apps do as well) include all of the information from the chart supplement in various fields and tabs (including tower frequencies and hours of operation) in their airport information page. Foreflight also includes the actual page from the chart supplement page as well, just in case you want to look at the original source. So there is no need buy or download the chart supplement separately, but you do need check that information in your flight planning app of choice before departing.