Alan Stibbs (see here) suggests some ways that the administration of the Lord's Supper can be made more biblical and thus convey more clearly its function and meaning. I'm not sure what I think about every point he makes, but it ought to spark discussion at the least (italics and bold bits are my emphases).
Firstly he says 'Blessing God the Giver is the proper way to consecrate material things for men's use. So new extended thanksgivings are desirable, first, for the bread, and later for the wine, similar to those regularly offered in some Free Church forms of service...these thanksgivings should be regarded as the consecration of the bread and wine for their use; without any introduction at this point of the decisive words which indicate their sacramental significance.'
Secondly, 'our Lord's declaratory words, "This is my body given for you", "This is my blood shed for many", should be removed from the introductory consecration, and, in accordance with the pattern of the Lord's institution, made an essential and simultaneous part of the actual administration.' This, he argues, is because the 'words and actions together of the movement of administration' make the bread and wine sacramental. Hence to disjoin the words from the action is to half do the job, or imply that the sacrament exists apart from the administration.
The third recommended change is that 'the bread and wine ought deliberately to be kept apart and administered separately, first the bread to all, and later the cup to all.' The reason is to 'fully follow the pattern of our Lord's institution, and to preserve the vivid winess to His death which we thus dramatically remember' and also 'make fellowship with others' easier (i.e. evangelical Free Church bros and sisters who already administer the bread and wine separately).
Having suggested these three changes to the CofE order of service, Stibbs goes on to suggest other areas where there is 'room and urgent need for a fuller obedience to the teaching and the principles of God's written Word' regarding the Supper. These include -
a. Administration of the supper 'by any member whom the body of believers may entrust with this ministry'. Having noted that things need to be orderly, and that giving this ministry to entrusted elders fits in with this sense of order, he also asks 'why, for the lack of a bishop or presbyter, should congregations be deprived of the Lord's Supper, when they have in theri midst mature and godly members, who could, if given the opportunity, worthily fulfil the necessary ministry?'
b. Not having 'a so-called sanctuary at the East end' of a church building that suggests the sacrament is the 'exclusive preserve of a special ministry'. Stibbs then advocates bringing the table into the middle of the congregation and that before communion begins the church should be 'conveniently placed for the reception of the sacrament, without further movement on their part'. This would restore the supper to what he sees it was intended to be in the rubrics - 'a corporate act in which a number share' with every believer 'directly at or around the Lord's table throughout the whole service'.
[All quotes from p84-88 of 'Sacrament, Sacrifice and Eucharist' by Alan Stibbs (London: Tyndale, 1961).]