One thing that can be done, you could create a new specific handler inheriting from TextOutputHandler, and set the filename you want. For example:
type
TTMSLoggerDailyTextOutputHandler = class(TTMSLoggerTextOutputHandler)
protected
function GetOutputFile: string; override;
end;
procedure TForm4.FormCreate(Sender: TObject);
begin
TMSDefaultLogger.RegisterOutputHandlerClass(TTMSLoggerDailyTextOutputHandler,
['D:\trash\log\mylogfile__%date%.txt']);
end;
{ TTMSLoggingDailyOutputHandler }
function TTMSLoggerDailyTextOutputHandler.GetOutputFile: string;
begin
Result := StringReplace(FileName, '%date%', FormatDateTime(' mm-dd-yy-hhmm', Now), [rfReplaceAll]);
end;